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11.198 The base of a right prism is a triangle with sides 6, 8, and 10 cm. A certain plane section of this prism cuts off segments of 12 cm each from the lateral edges passing through the vertices of the larger and medium angles of the base, and a segment of 18 cm from the edge passing through the vertex of the smalles... | Solution.
Let $A B C A_{1} B_{1} C_{1}$ be the given right prism with base $A B C, A C=6, B C=8, A B=10$. Let $L M N$ be the given section, $A M=C N=12, B L=18$.
By the condition, $A B^{2}... | 336 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,113 |
11.199 The edge of a slanted parallelepiped is equal to $l$. To it, two adjacent faces are attached, with areas equal to $m^{2}$ and $n^{2}$, and their planes form an angle of $30^{\circ}$. Calculate the volume of the parallelepiped. | ## Solution.
$B_{I}$
Let $T=A B C D A_{1} B_{1} C_{1} D_{1}$ be the given pa-
rallelepiped with base $A B C D$,
$$
S_{A A_{1} D_{1} D}=m^{2}, S_{C C_{1} D_{1} D}=n^{2}, D D_{1}=l, \text {... | \frac{^{2}n^{2}}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,114 |
11.200 Through a point dividing the edge of a regular tetrahedron in the ratio 1 : 4, a plane perpendicular to this edge is drawn. Find the ratio of the volumes of the resulting parts of the tetrahedron. | ## Solution.
Let $A B C D$ be the given regular tetrahedron, $A K: K B=4: 1$ and the section $K L M \perp A B$. Let $E$ be the midpoint of $A B$. Then $K L M \| E C D$, as planes with paral... | \frac{4}{121} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,115 |
11.201 The lateral edges of a triangular pyramid are of equal length and equal to $a$. Of the three planar angles formed by these edges at the apex of the pyramid, two contain $45^{\circ}$, and the third contains $60^{\circ}$. Determine the volume of the pyramid. | ## Solution.
Let $FABC$ be the given pyramid with base $ABC$, $\angle AFC = 60^\circ$, $\angle BFC = \angle AFB = 45^\circ$. Drop perpendiculars from $A$ and $C$ to the edge $FB$. They inte... | \frac{^3}{12} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,116 |
### 11.203 Through every three vertices of a cube, located at the ends
of each triplet of edges converging at one vertex, a plane is drawn. Find the volume of the body bounded by these planes, if the edge of the cube is $a$. | ## Solution.
A
We connect the center of each face of the given cube with the centers of all adjacent faces. The resulting body $K N L Q M P=T$ is the desired one. To verify this, it is suf... | \frac{^{3}}{6} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,117 |
11.205 Two regular tetrahedra are joined by two faces to form a double pyramid. The centers of the six lateral faces of this double pyramid are taken as the vertices of a right triangular prism. Calculate the volume of this prism if the edge of the tetrahedron is $a$. | Solution.
Let $M, K, N$ be the centers of three faces of the given double pyramid $F F^{\prime} A B C$, and $O$ be the center of $\triangle A B C$. Let $P$ and $D$ be the midpoints of $A C$... | \frac{\sqrt{2}^{3}}{54} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,118 |
### 11.206 The side of the base of a regular triangular pyramid is equal to
$a$; the area of its section, which has the shape of a square, is equal to $m^{2}$. Find the ratio of the lateral surface area of the pyramid to the area of the base. | ## Solution.
Let $K L M N$ be the section of the given regular pyramid $F A B C$, which is a square. $M N \| L K \Rightarrow$ $M N\|F B C \Rightarrow M N\| F C$ (Theorem 14). For the same ... | \frac{\sqrt{9^{2}-3^{2}+6}}{-} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,119 |
11.207 Two cubes with an edge equal to $a$ have a common segment connecting the centers of two opposite faces, but one cube is rotated by $45^{\circ}$ relative to the other. Find the volume of the common part of these cubes. | ## Solution.
Let $E$ and $F$ be the centers of the faces $ABCD$ and $KLMN$ of the given cube $T_1 = ABCD KLMN$, and $T_2$ be the cube obtained from $T_1$ by rotating it $45^\circ$ around $E... | 2a^3(\sqrt{2}-1) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,120 |
11.208 Through the ends of three edges emanating from vertices $B, D, A_{1}$ and $C_{1}$ of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$, with edge length $a$, planes are drawn. Prove that the resulting figure is a regular tetrahedron, and calculate its total surface area and volume. | Solution.
The desired tetrahedron \( T = A B_{1} C D_{1} \) is regular, as each of its faces is an equilateral triangle with side length \( a \sqrt{2} \) (the diagonals of the faces of the ... | 2\sqrt{3}^{2};\frac{^{3}}{3} | Geometry | proof | Yes | Yes | olympiads | false | 52,121 |
11.209 A plane is drawn through the side of the base of a regular quadrilateral pyramid, which cuts off a triangle with an area of $4 \mathrm{~cm}^{2}$ from the opposite face. Find the lateral surface of the pyramid that is cut off by the drawn plane from the given pyramid, if the lateral surface of the given pyramid i... | ## Solution.
Let the section $C D M N$ of the given regular pyramid $F A B C D$ cut off a lateral face $\triangle F M N$ with an area of 4. $C D\|A B \Rightarrow M N\| A B$ (Theorem 14) $\R... | 20.25\mathrm{~}^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,122 |
11.210 Prove that the volumes of two triangular pyramids, each having an equal trihedral angle, are in the ratio of the products of the lengths of the three edges of the equal trihedral angles. | Solution. Let the pyramids $FABC, F_1A_1B_1C_1$ have the same trihedral angles at vertices $F$ and $F_1$. Drop the height $AO$ onto the face $FBC$. Then
$AO = AF \cdot \sin (\angle AFO)$ and $V_{FABC} = \frac{1}{3} AO \cdot S_{FBC} = \frac{1}{3} AF \sin (\angle AFO) \cdot BF \cdot CF \cdot \sin (\angle BFC)$.
Similarl... | proof | Geometry | proof | Yes | Yes | olympiads | false | 52,123 |
### 11.211 The side of the base of a regular quadrilateral pyramid
is equal to $a$, the lateral edge forms an angle of $30^{\circ}$ with the height. A plane is drawn through the vertex of the base of the pyramid, perpendicular to the opposite lateral edge. This plane divides the pyramid into two parts. Determine the v... | ## Solution.
Let $F A B C D$ be the given regular pyramid, and the section $D M K N$ is perpendicular to $B F$.
By the condition $\angle B F D=60^{\circ} \Rightarrow \triangle B F D$ is an... | \frac{\sqrt{6}^{3}}{18} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,124 |
11.212 The distance between non-intersecting diagonals of two adjacent lateral faces of a cube is $d$. Determine the total surface area of the cube. | ## Solution.
Let the distance between the diagonals $C N$ and $B D$ of the faces of the given cube $A B C D K L M N$ be $d$. $\triangle B D K \| \triangle C L N$, since $L C\|K D, C N\| B K... | 18^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,125 |
11.213 Calculate the volume of a triangular pyramid, in which two opposite edges are 4 and 12 cm, and each of the other edges is 7 cm. | ## Solution.
Let in the given pyramid $F A B C$, $F B=12, A C=4$, and the other edges are equal to 7. If $K$ is the midpoint of $A C$, then $F K$ and $B K$ are the heights of the isosceles... | 24^3 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,126 |
11.214 The faces of the parallelepiped are rhombuses, the diagonals of which are 3 and 4 cm. The parallelepiped has trihedral angles composed of three acute angles of the rhombuses. Find the volume of the parallelepiped. | ## Solution.
Let $A$ be the vertex of the given parallelepiped $T=A B C D A_{1} B_{1} C_{1} D_{1}$, to which the acute angles of its faces adjoin.
Since a rhombus is uniquely determined by... | \frac{9\sqrt{39}}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,127 |
11.215 Find the volume of a triangular pyramid, the sides of the base of which are equal to $a, b$ and $c$, if each of these sides is equal to the lateral edge that does not intersect with it. | ## Solution.
Through each edge of the given pyramid $A B C D$, we draw a plane parallel to the opposite edge. The resulting body $T=A N D K L B M C$ is a parallelepiped, and $V_{A B C D}=\... | \frac{1}{3}\sqrt{\frac{^{2}+b^{2}-^{2}}{2}}\sqrt{\frac{^{2}+^{2}-b^{2}}{2}}\sqrt{\frac{b^{2}+^{2}-^{2}}{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,128 |
11.217 The polyhedron has the following structure: two of its faces (bases) are polygons located in parallel planes; the other faces (lateral) are trapezoids, parallelograms, or triangles, each vertex of which is simultaneously a vertex of one of the bases. Prove that the volume of such a polyhedron is $\frac{1}{6} H \... | Solution. The given polyhedron can be divided into tetrahedra \( T \), the vertices of which coincide with the vertices of the polyhedron. Therefore, by proving the required formula for an arbitrary tetrahedron \( T \) and summing their volumes, we will obtain the same formula for the entire polyhedron. There are two c... | proof | Geometry | proof | Yes | Yes | olympiads | false | 52,129 |
11.219 The diagonals of two identical cubes with an edge equal to $a$ lie on the same straight line. A vertex of the second cube coincides with the center of the first, and the second cube is rotated around the diagonal by $60^{\circ}$ relative to the first. Find the volume of the common part of these cubes. | ## Solution.
Let the first cube be denoted as $T_{1}$, the second cube as $T_{2}=A B C D K L M N$, and after rotation, the second cube as $T_{2}^{n}$. The common part of the cubes $T_{1}$ a... | \frac{9^{3}}{64} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,130 |
11.220 A truncated cone is described around a sphere, the area of the lower base of which is a times larger than the area of its upper base. How many times is the volume of the truncated cone larger than the volume of the sphere? | Solution.
The axial section of the given truncated cone is an isosceles trapezoid \(ABCD\), in which a circle with center \(O\) and radius \(r\) can be inscribed, where \(r\) is the radius ... | \frac{\sqrt{}+1}{2\sqrt{}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,131 |
11.221. A sphere is inscribed in a cone. Prove that the ratio of the total surface area of the cone to the surface area of the sphere is equal to the ratio of their volumes. | Solution. Let $l$ be the slant height of the cone, $r$ be the radius of its base, and $R$ be the radius of the sphere inscribed in the cone. Then the axial section of the cone is an isosceles $\triangle ABC$ with base $2r$ and side $l$, and $R$ is the radius of the inscribed circle $\Rightarrow S_{ABC}=R(l+r)=h \cdot r... | proof | Geometry | proof | Yes | Yes | olympiads | false | 52,132 |
### 11.222 The height of the cylinder is equal to the radius of its base and has a length
a. Through the axis of the cylinder, another cylindrical surface is drawn, dividing the circumference of the base into two arcs, the lengths of which are in the ratio 2:1. This cylindrical surface divides the given cylinder into... | ## Solution.
Let the base of the given cylinder, a circle with center \(O_{1}\), be divided by another circle passing through \(O_{1}\) into two arcs
\(\cup A C B, \cup A D B\), the length... | 2\pi^{2};\frac{^{3}}{6}(2\pi+3\sqrt{3}) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,133 |
11.225 A cone lies on a plane and rolls along it, rotating around its fixed vertex. The height of the cone and its slant height are equal to $h$ and $l$. Calculate the area of the surface described by the height of the cone. | ## Solution.
Let $\triangle A B F$ be the axial section of the given cone,
$l=A F=B F$. Since the angle $A F K$ between the height $K F$ of the given cone and the plane on which it rolls doe... | \frac{\pi^{3}}{} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,134 |
11.226 The base of the pyramid $SABC$ is a triangle $ABC$ such that $AB=AC=10$ cm and $BC=12$ cm. The face $SBC$ is perpendicular to the base and $SB=SC$. Calculate the radius of the sphere inscribed in the pyramid if the height of the pyramid is 1.4 cm. | ## Solution.
The height $S K$ of the given pyramid $S A B C$ lies in the plane of the face $S B C$, since $S B C \perp A B C$ (by the condition). Therefore, $K$ is the midpoint of $B C$ (tr... | \frac{12}{19} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,135 |
11.227 The lengths of the lateral edges of a triangular pyramid are equal to $a, b$, and $c$; the planar angles formed by these edges are right angles. Find the length of the height dropped to the base of the pyramid. | ## Solution.
Let in the given pyramid $FABC$ the lateral edges $FA, FB, FC$ be pairwise perpendicular and equal to $a, b$, and $c$ respectively. Draw the height $AK$ in $\triangle ABC$. $FA... | \frac{abc}{\sqrt{^2b^2+^2^2+b^2^2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,136 |
11.228 Prove that if a sphere can be inscribed in a polyhedron, then its volume is $1 / 3$ of the product of the total surface area of the polyhedron and the radius of the inscribed sphere. | ## Solution.
Connect the center $O$ of the inscribed sphere with the vertices of the polyhedron $T$. Then $T$ will be divided into pyramids with vertices $O$, bases of which are the faces of $T$, and heights equal to the radius $r$ of the sphere. Then
$V_{T}=\frac{1}{3} r \cdot S_{1}+\frac{1}{3} r \cdot S_{2}+\ldots+... | proof | Geometry | proof | Yes | Yes | olympiads | false | 52,137 |
11.230 The height of a regular triangular pyramid is equal to $\boldsymbol{H}$. Find its total surface area if a plane, passing through the vertex of the base of the pyramid perpendicular to the height of the opposite lateral face, forms an angle of $30^{\circ}$ with the plane of the axis. | ## Solution.
Let in the given regular pyramid $F A B C$, the height $F L$ of the lateral face $F A C$ be perpendicular to the section $B M N$. Since $M N \| A C$ and $M N \perp F L$, the li... | 3\sqrt{3}H^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,139 |
11.231 The base of the pyramid $S A B C$ is a right isosceles triangle $A B C$, with the length of the hypotenuse $A B=4 \sqrt{2}$. The lateral edge $S C$ of the pyramid is perpendicular to the plane of the base, and its length is 2. Find the measure of the angle and the distance between the lines, one of which passes ... | Solution.
Let $K$ and $M$ be the midpoints of edges $AC$ and $AB$, and $N$ be the midpoint of segment $AM$.
$CM \| KN \Rightarrow CM \| SKN \Rightarrow$ the distance $x$ between $CM$ and ... | \frac{\pi}{3};\frac{2\sqrt{3}}{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,140 |
11.232 Prove that if a tetrahedron is orthocentric, i.e., such that the lines containing its altitudes intersect at one point, then: a) each pair of opposite edges are mutually perpendicular; b) if one of the dihedral angles at any vertex of the tetrahedron is a right angle, then the other two dihedral angles are also ... | Solution.
Let $ABCD$ be an orthocentric tetrahedron, and $O$ be the point of intersection of its altitudes.
a) $BC \perp DO, BC \perp AO \Rightarrow BC \perp AOD$ (Theorem 11) $\Rightarrow B... | proof | Geometry | proof | Yes | Yes | olympiads | false | 52,141 |
11.233 a) The lengths of the edges $AB, AC, AD$ and $BC$ of an orthocentric tetrahedron are 5, 7, 8, and 6 cm, respectively. Find the lengths of the remaining two edges. b) Is the tetrahedron $ABCD$ orthocentric if $AB=8$ cm, $BC=12$ cm, and $DC=6$ cm? | Solution. a) According to item v) of problem 11.232, $A B^{2}+C D^{2}=A C^{2}+B D^{2}=A D^{2}+B C^{2}=100 \Rightarrow C D=\sqrt{75}, B D=\sqrt{51}$.
Answer: $5 \sqrt{3}$ cm, $\sqrt{51}$.
b) If the tetrahedron were orthocentric, then
$A B^{2}+C D^{2}=100=B C^{2}+A D^{2} \Rightarrow A D^{2}=100-144<0$, which is imposs... | 5\sqrt{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,142 |
11.234 In an orthocentric tetrahedron $ABCD$, the angle $ADC$ is right. Prove that $\frac{1}{h^{2}}=\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}$, where $h$ is the length of the height of the tetrahedron drawn from vertex $D$, $a=DA$, $b=DB$, $c=DC$. | Solution. According to item b) of problem 11.232, the edges $A D, B D, C D$ are pairwise perpendicular. The conditions of problem 11.227 are satisfied, from the solution of which it follows that $h=\frac{a b c}{\sqrt{a^{2} b^{2}+b^{2} c^{2}+a^{2} c^{2}}} \Rightarrow \frac{1}{h}=\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}+\fr... | \frac{1}{}=\sqrt{\frac{1}{^{2}}+\frac{1}{b^{2}}+\frac{1}{^{2}}} | Geometry | proof | Yes | Yes | olympiads | false | 52,143 |
11.235 In an orthocentric tetrahedron $A B C D$, the angle $A B C$ is right, $S_{1}, S_{2}, S_{3}$ are the areas of the faces $B A C, B A D, B C D$ respectively. Prove that the volume of the tetrahedron is $\frac{1}{3} \sqrt{2 S_{1} S_{2} S_{3}}$. | Solution. According to item b) of problem 11.232, the edges $B A, B C, B D$ are pairwise perpendicular. Let $a=B A, b=B C, c=B D$.
Then $V_{A B C D}=\frac{a b c}{6}=\frac{1}{6} \sqrt{(a b) \cdot(b c) \cdot(a c)}=\frac{1}{3} \sqrt{2 S_{1} S_{2} S_{3}}$, which is what we needed to prove.
## GEOMETRY PROBLEMS INVOLVING ... | \frac{1}{3}\sqrt{2S_{1}S_{2}S_{3}} | Geometry | proof | Yes | Yes | olympiads | false | 52,144 |
12.391 The sides of the parallelogram are equal to $a$ and $b(a<b)$. The smaller diagonal forms an obtuse angle with the smaller side and an angle $\alpha$ with the larger side. Find the larger diagonal of the parallelogram. | ## Solution.
B
Let in the given parallelogram $ABCD$, $a=AB, b=BC, \alpha=\angle ADB, \beta=\angle BDC$. By the Law of Sines for $\triangle ABD$,
$$
\begin{aligned}
& \frac{a}{\sin \alpha}... | \sqrt{^{2}+b^{2}+2b(\cos\alpha\cdot\sqrt{^{2}-b^{2}\sin^{2}\alpha}+b\cdot\sin^{2}\alpha)} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,145 |
12.392 In the sector $P O Q$ of radius $R$ with a central angle $\alpha$, a rectangle is inscribed; two of its vertices lie on the arc of the sector, and the other two on the radii $P O$ and $P Q$. Find the area of the rectangle, if the acute angle between its diagonals is $\beta$. | ## Solution.
Let the rectangle $ABCD$ be inscribed in the given sector $POQ$, and let $N$ be the midpoint of side $AD$. Then $ON \perp AD$ (as a radius passing through the midpoint of a cho... | \frac{4R^2\cdot\operatorname{tg}\frac{\beta}{2}}{1+(\operatorname{ctg}\frac{\alpha}{2}+2\operatorname{tg}\frac{\beta}{2})^2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,146 |
12.393 In triangle $A B C$, the acute angles $\alpha$ and $\gamma (\alpha>\gamma)$ at the base $A C$ are given. From vertex $B$, the altitude $B D$ and the angle bisector $B E$ are drawn. Find the area of triangle $B D E$, if the area of triangle $A B C$ is $S$. | Solution.
$$
\begin{aligned}
& \text { B } \\
& \text { Let } \alpha=\angle A, \gamma=\angle C \text {. Then } \\
& \text { } \\
& \angle A B D<\angle D B C \Rightarrow \text { the bisector } B E \text { lies inside } \\
& \triangle D B C . \angle D B E=\angle A B E-\angle A B D= \\
& =\frac{180^{\circ}-(\alpha+\gamma... | S\cdot\tan(\frac{\alpha-\gamma}{2})\cdot\frac{\sin\alpha\cdot\sin\gamma}{\sin(\alpha+\gamma)} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,147 |
12.394 Two equal circles are inscribed in a segment of a circle of radius $R$, touching each other, the arc of the segment, and its chord. Find the radii of these circles if the central angle subtended by the arc of the segment is $\alpha(\alpha<\pi)$. | ## Solution.
Let $P_{1}$ and $P_{2}$ be the centers of the circles of radius $x$ inscribed in the segment $A B$ of the given circle with center $O$, and $L_{1}$ and $L_{2}$ be the points of ... | 4R\cdot\cos\frac{\alpha}{4}\cdot\sin^{2}\frac{\alpha}{8} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,148 |
12.395 The ratio of the radius of the circle inscribed in an isosceles triangle to the radius of the circle circumscribed around it is $\boldsymbol{m}$. Find the angles of the triangle and the permissible values of $\boldsymbol{m}$. | ## Solution.
Let $O$ be the center of the inscribed circle of radius $r$ in the given $\triangle ABC$, $\alpha = \angle A = \angle C$, and $R$ be the radius of the circumcircle of $\triang... | \arccos\frac{1+\sqrt{1-2m}}{2},2\arcsin\frac{1+\sqrt{1-2m}}{2}or\arccos\frac{1-\sqrt{1-2m}}{2},2\arcsin\frac{1-\sqrt{1-2m}}{2};0< | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,149 |
12.396 In a parallelogram, two sides $a$ and $b(a>b)$ and the acute angle $\alpha$ between the diagonals are given. Find the angles of the parallelogram. | ## Solution.
$$
\begin{aligned}
& A \\
& \Rightarrow 2 \cdot A \check{C} \cdot O K=a^{2}-b^{2} \Rightarrow S_{A B C D}=B K \quad A C= \\
& =O K \cdot \tan \alpha \cdot A C=\frac{a^{2}-b^{2... | \arcsin(\frac{^{2}-b^{2}}{2}\tan\alpha),\pi-\arcsin(\frac{^{2}-b^{2}}{2}\tan\alpha) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,150 |
12.397 In a segment with a central angle $\alpha$, a regular triangle is inscribed such that one of its vertices coincides with the midpoint of the segment's chord, and the other two lie on the arc of the segment. The height of the triangle is $h$. Find the radius of the arc of the segment. | ## Solution.
284
Let $O$ be the center of the circle with radius $x$ and the given segment $P Q, \alpha=\angle P O Q, B$ be the midpoint of the chord $P Q, \triangle A B C$ be an equilater... | \frac{}{\sin^{2}\frac{\alpha}{2}}\cdot(\cos\frac{\alpha}{2}+\sqrt{1+\frac{1}{3}\sin^{2}\frac{\alpha}{2}}) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,151 |
### 12.398 The distance between the centers of two externally tangent circles is $d$. The angle between their common external tangents is $\alpha$ radians. Find the area of the curvilinear triangle bounded by a segment of one tangent and the two corresponding arcs of the circles. | ## Solution.
Let $O$ and $P$ be the centers of two given circles with radii $x$ and $y$, touching at point $C$, and $AB$ be their external tangent. Let $PK$ be the height of the trapezoid $A... | S_{ACB}=\frac{^{2}}{8}(4\cos\frac{\alpha}{2}-\pi(1+\sin^{2}\frac{\alpha}{2})+2\alpha\cdot\sin\frac{\alpha}{2}) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,152 |
12.399 In a parallelogram, the two sides are $\boldsymbol{a}$ and $\boldsymbol{b}(\boldsymbol{a}>\boldsymbol{b})$ and the height $h$, drawn to the larger side. Find the acute angle between the diagonals of the parallelogram. | ## Solution.
$$
\begin{aligned}
& \text { } \\
& \text { Let } ABCD \text { be the given parallelogram, } \\
& =a^{2}-\left(\frac{AC}{2}+OK\right)^{2} \Rightarrow 2 \cdot AC \cdot OK=a^{2}-b^{2} \Rightarrow \\
& \Rightarrow S_{ABCD}=AC \cdot BK=AC \cdot OK \cdot \operatorname{tg} \alpha=\frac{a^{2}-b^{2}}{2} \cdot \op... | \operatorname{arctg}\frac{2}{^{2}-b^{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,153 |
12.401 The sides of a parallelogram are equal to $a$ and $\boldsymbol{b}(\boldsymbol{a}<\boldsymbol{b}$). From the midpoint of the larger side, the parallel side is seen at an angle $\alpha$. Find the area of the parallelogram. | Solution.
On the other hand, $B K^{2}+C K^{2}=b^{2}+2 \cdot B K \cdot C K \cdot \cos \alpha$ (cosine theorem for $\triangle B C K) \Rightarrow B K \cdot C K=\frac{4 a^{2}-b^{2}}{4 \cos \al... | \frac{4^{2}-b^{2}}{4}\cdot\tan\alpha | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,155 |
12.402 In a triangle, two sides $a$ and $b(a>b)$ and the area $S$ are given. Find the angle between the height and the median drawn to the third side. | ## Solution.
$$
\begin{aligned}
& \text { Let in the given } \triangle A B C \quad a=B C, \dot{b}=A B, B D \text { - altitude, } \\
& B M-\text { median of } \triangle A B C . \text { Then } \\
& B D^{2}=A B^{2}-\left(\frac{A C}{2}-D M\right)^{2}=B C^{2}-\left(\frac{A C}{2}+D M\right)^{2} \Rightarrow \\
& \Rightarrow ... | \operatorname{arctg}\frac{^{2}-b^{2}}{4S} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,156 |
12.403 The ratio of the radius of the circle circumscribed around a trapezoid to the radius of the circle inscribed in it is $\boldsymbol{k}$. Find the angles of the trapezoid and the permissible values of $k$. | Solution.
Let $r$ be the radius of the circle inscribed in the given trapezoid $ABCD (BC \| AD)$, $R$ be the radius of the circle circumscribed around $ABCD$, $CK$ be the height of the trap... | \arcsin\frac{1}{k}\sqrt{\frac{1+\sqrt{1+4k^2}}{2}},\pi-\arcsin\frac{1}{k}\sqrt{\frac{1+\sqrt{1+4k^2}}{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,157 |
12.404 The ratio of the perimeter of a parallelogram to its larger diagonal is $k$. Find the angles of the parallelogram, given that the larger diagonal divides one of the angles of the parallelogram in the ratio 1:2. | ## Solution.
Let $AC$ be the longer diagonal of the given parallelogram $ABCD$, $\angle CAD = x$, and $\angle BAC = 2x$. Extend side $AD$ by a segment $KD$ equal to $CD$.
$$
\angle AKC = ... | 3\arccos\frac{2+k}{2k},\pi-3\arccos\frac{2+k}{2k} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,158 |
12.405 In an isosceles triangle $A B C$, an equilateral triangle $D E F$ is inscribed; point $D$ lies on side $B C$, point $E$ - on side $A C$, and point $F$ - on side $A B$. The side $A B$ is to the side $D F$ as 8:5. Find the sine of angle $D E C$. | ## Solution.
Let $\alpha=\angle D E C$. Then $\angle E D C=120^{\circ}-\alpha$, $\angle F D B=180^{\circ}-\left(60^{\circ}+\angle E D C\right)=\alpha \Rightarrow$ $\Rightarrow \triangle E D ... | \frac{4\sqrt{3}\3}{10} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,159 |
12.406 The tangent of the angle between the median and the altitude drawn to the lateral side of an isosceles triangle is $\frac{1}{2}$. Find the sine of the angle at the vertex. | ## Solution.
$\boldsymbol{B}$
A
Let $A M$ and $A D$ be the median and altitude of the given $\triangle A B C, A B=B C$. Then $\sin \angle B=\frac{A D}{A B}$, $A B^{2}=A D^{2}+B D^{2}, B D... | \frac{3}{5} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,160 |
12.407 A line perpendicular to the chord of a segment divides the chord in the ratio 1:4, and the arc - in the ratio $1: 2$. Find the cosine of the central angle subtended by this arc. | ## Solution.
Let $O$ be the center of the circle with radius $R$ containing the segment $AB$, such that $\cup AN : \cup NB = 1:2$, $AM : MB = 1:4$, and $MN \perp AB$. Draw the radius $OK \p... | -\frac{23}{27} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,161 |
12.408 In an acute-angled triangle $ABC$ with side $AC$ equal to $\boldsymbol{b}, \angle \boldsymbol{A}=\alpha, \angle B=\beta$ radians. A circle is drawn through the orthocenter (the point of intersection of the altitudes) and the feet of the altitudes dropped onto sides $AB$ and $BC$. Find the area of the common part... | Solution. Let $P$ be the orthocenter of the given $\triangle ABC$, and $AE, CD$ be its altitudes.
The circle passing through points $P, E, D$ will also pass through vertex $B$ (Theorem 8). By the same Theorem 8, $BP$ is the diameter of this circle ($\angle BDP = 90^\circ$).
) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,162 |
12.409 In an acute-angled triangle $ABC$, the angles are known. Find the ratio in which the orthocenter (the point of intersection of the altitudes) divides the altitude drawn from vertex $A$. | Solution.
Let $O$ be the orthocenter of the given $\triangle ABC$,
$AD$ - its altitude. Then $S_{OBD} = \frac{BD \cdot BO}{26} \cdot \sin (\angle OBD)$,
$$
S_{OBA} = \frac{AB \cdot BO}{2... | \frac{\cos\angleA}{\cos\angleB\cdot\cos\angleC} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,163 |
12.410 In an acute-angled triangle $A B C$, altitudes $A L$ and $C N$ are drawn. Find the radius of the circle passing through points $B, L$, and $N$, if $A C=a$ and $\angle A B C=\alpha$. | ## Solution.
Let $O$ be the orthocenter of $\triangle ABC$. According to Theorem 8, the circle passing through points $B, N, L$ will also pass through $O$. Moreover, by Theorem 6, $BO$ is it... | \frac{1}{2}\cdot\cot\alpha | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,164 |
12.411 Prove that the ratio of the area of any triangle to the area of the circle circumscribed around it is less than $2 / 3$.
---
The text has been translated while preserving the original formatting and line breaks. | Solution. Let the sides and angles of $\triangle A B C$ be $a, b, c$ and $\alpha, \beta, \gamma ; R$-the radius of the circumscribed circle. Then $S_{A B C}=\frac{1}{2} \cdot b \cdot c \cdot \sin \alpha=2 R^{2} \sin \alpha \cdot \sin \beta \cdot \sin \gamma$ (by the sine theorem) $\Rightarrow \frac{S_{A B C}}{\pi R^{2}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 52,165 |
12.413 The lengths of the four arcs into which the entire circumference of radius $R$ is divided form a geometric progression with a common ratio of 3. The points of division serve as the vertices of a quadrilateral inscribed in this circle. Find its area. | Solution. Let points $A, B, C, D$ divide the given circle with center $O$ and radius $R$ such that $\cup A B: \cup B C: \cup C D: \cup D A=1: 3: 9: 27$.
Let $\alpha=\angle A O B$. Then $\angle B O C=3 \alpha, \angle C O D=9 \alpha, \angle D O A=27 \alpha \Rightarrow \alpha=9^{\circ}$.
$$
\begin{gathered}
S_{A B C D}=... | \frac{R^{2}\sqrt{2}}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,166 |
12.414 One of the plane angles of a trihedral angle is equal to $\alpha$. The dihedral angles adjacent to this plane angle are equal to $\beta$ and $\gamma$. Find the other two plane angles. | ## Solution.
Let the plane angle $\angle E_{2} F E_{3}$ of the given trihedral angle $F E_{1} E_{2} E_{3}$ be $\alpha$. From an arbitrary point $A$ on the ray $F E_{1}$, drop perpendiculars... | \operatorname{arctg}(\operatorname{ctg}\alpha\cdot\sin\beta(\operatorname{ctg}\beta+\frac{\operatorname{ctg}\gamma}{\cos\alpha}));\operatorname{arctg}(\operatorname{ctg}\alpha\cdot\sin\gamma(\operatorname{ctg}\gamma+\frac{\operatorname{} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,167 |
12.416 A sphere is inscribed in a cone such that their surfaces touch. The volume of the body enclosed between them is 8 times smaller than the volume of the sphere. Find the angle at the vertex of the axial section of the cone. | ## Solution.
Thus, applying known formulas for volume calculations, we have:
$$
V_{C}=\frac{2}{3} \pi R^{2} \cdot M K, V_{T_{1}}=\frac{1}{3} \pi \cdot A K^{2} \cdot F K, \quad V_{T_{2}}=\f... | \frac{\pi}{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,169 |
12.417 A sphere is inscribed in a regular quadrilateral pyramid. A tangent plane parallel to the base of the pyramid is drawn to the sphere, dividing the volume of the pyramid in the ratio $m: n$, counting from the vertex. Find the angle between the height of the pyramid and its lateral face. | Solution. Let $A$ and $D$ be the midpoints of the edges $M E$ and $N L$ of the base of the given regular pyramid FEMNL, $B C$ be the line of intersection of $A F D$ with the given plane tangent to the sphere inscribed in the pyramid, $O$ be the projection of the vertex $F$ onto the base.
$N L \perp A D \Rightarrow N L... | \arcsin\frac{\sqrt[3]{+n}-\sqrt[3]{}}{\sqrt[3]{+n}+\sqrt[3]{}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,170 |
12.418 Through the vertex of the base of a regular triangular pyramid, a plane is drawn perpendicular to the opposite lateral face and parallel to the opposite side of the base. This plane forms an angle $\alpha$ with the plane of the pyramid's base. Find the plane angle at the vertex of the pyramid. | ## Solution.
Let $FABC$ be the given regular pyramid, $O$ be the center of the base, and the plane $AMN \perp FBC$. By the condition, $AMN \| BC \Rightarrow MN \| BC$ (Theorem 14). The lin... | 2\arctan(\sqrt{3}\cdot\sin\alpha) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,171 |
12.419 A rectangle rotates around an axis passing through its vertex parallel to the diagonal. Find the surface area of the solid of revolution, if the area of the rectangle is $S$, and the angle between the diagonals is $\alpha$. | ## Solution.
Let \(ABCD\) be the given rectangle rotating around the line \(l\), where \(l \parallel BD\). The surface of the solid of revolution \(T\) consists of the surfaces of two cones... | 4\sqrt{2}\piS\cdot\sin(\frac{\alpha}{2}+\frac{\pi}{4}) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,172 |
12.420 Find the radius of the sphere that touches the base and the lateral edges of a regular triangular pyramid, where the side of the base is equal to $a$, and the dihedral angle at the base is equal to $\alpha$. | Solution.
Let $O$ be the center of the sphere that touches the lateral edges $F A, F B, F C$ and the base of the given regular pyramid $F A B C$ at points $M, N, K$ and $P$ respectively. $\... | \frac{\sqrt{3}}{3}(-2\operatorname{ctg}\alpha+\sqrt{4\operatorname{ctg}^{2}\alpha+1}) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,173 |
12.422 A regular triangular pyramid is intersected by a plane passing through its lateral edge and height. In the section, a triangle with an angle $\pi / 4$ at the vertex of the pyramid is formed. Find the angle between the lateral edge and the base plane of the pyramid. | ## Solution.
Let $F P$ be the height of the given regular pyramid $F A B C, K$ be the point of intersection of the plane $F C P$ with the edge $A B$. $K$ is the midpoint of $A B$ ($C K$ pas... | \operatorname{arctg}\frac{3+\sqrt{17}}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,174 |
12.423 The side of the base of a regular triangular pyramid is equal to $a$, and the plane angle at the vertex is $\alpha$. A sphere is inscribed in the pyramid. Find the area of the section of this sphere by a plane passing through the center of the base and perpendicular to its lateral edge. | ## Solution.
Let $F P$ be the height of the given regular pyramid $F A B C, O$ be the center of the inscribed sphere, and $K N M$ be the section of the pyramid by a plane passing through $... | \frac{\pi^{2}}{9}\cdot\sin^{2}\frac{\alpha}{2}\cdot\frac{\sin(\frac{\pi}{3}-\frac{\alpha}{2})}{\sin(\frac{\pi}{3}+\frac{\alpha}{2})} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,175 |
12.424 The base of a pyramid inscribed in a cone is a quadrilateral, one side of which is equal to $a$, and each of the other three sides is equal to $b$. The vertex of the pyramid lies at the midpoint of one of the generatrices. Find the volume of the pyramid if the angle between the generatrix and the height of the c... | ## Solution.
Let the base of the given pyramid $FABCD$ be a quadrilateral $ABCD$ inscribed in a circle of radius $R$, which is the base of the given cone, $AB = BC = CD = b, AD = a, P$ - th... | \frac{\cot\alpha}{24}(b^2+)^{\frac{3}{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,176 |
12.425 The ratio of the volume of a truncated cone to the volume of a sphere inscribed in it is $\boldsymbol{k}$. Find the angle between the slant height of the cone and the plane of its base and the permissible values of $k$. | ## Solution.
Let $A B C D$ be the axial section of the given truncated cone. Then $A B C D$ is an isosceles trapezoid, in which a circle of radius $R$ (where $R$ is the radius of the sphere inscribed in the given cone) can be inscribed.
Then, if $B C=2 r_{1}, A D=2 r_{2}, C K$ is the height of $A B C D$, then $C D=r_... | \operatorname{arctg}\frac{2}{\sqrt{2k-3}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,177 |
12.426 The axial section of the cylinder is a square. The segment $A B$, connecting point $\boldsymbol{A}$ on the circumference of the upper base with point $B$ on the circumference of the lower base of the cylinder, is equal to $a$ and is at a distance $b$ from the axis of the cylinder. Find the angle between the line... | ## Solution.
A
B
Let $OP$ be the axis of the given cylinder, $M$ and $N$ be the midpoints of $OP$ and $AB$, and $AC$ be the generatrix of the cylinder. In $\triangle ABC$, draw the line $N... | \frac{1}{2}\arccos(-\frac{4b^2}{^2}) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,178 |
12.427 A plane is drawn through the vertex of the base of a regular quadrilateral pyramid, intersecting the opposite lateral edge at a right angle. The area of the section is half the area of the base of the pyramid. Find the angle between the lateral edge and the base plane. | Solution.
Let $FO$ be the height of the given regular pyramid. In $\triangle ACF$, draw the height $AK$, then through the intersection point $T$ of it with the height $FO$, draw the segment... | \arcsin\frac{1+\sqrt{33}}{8} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,179 |
12.428 Given three pairwise mutually perpendicular rays $O M, O N$ and $OP$. On the ray $O M$, a point $A$ is taken at a distance $O A$, equal to $a$; on the rays $O N$ and $O P$, points $B$ and $C$ are taken, respectively, such that the angle $A B C$ is equal to $\alpha$, and the angle $A C B$ is equal to $\beta$. Fin... | ## Solution.
Let $x = OB, y = OC$. Then
$$
AB^2 = a^2 + x^2, \quad AC^2 = a^2 + y^2, \quad BC^2 = x^2 + y^2
$$
By the Law of Sines for $\triangle ABC$, we have:
$$
\begin{aligned}
& \fr... | \sqrt{-\cot\beta\cdot\tan(\alpha+\beta)},\quad\sqrt{-\cot\alpha\cdot\tan(\alpha+\beta)} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,180 |
12.429 A sphere is inscribed in a cone. The circle of contact between the spherical and conical surfaces divides the volume of the sphere in the ratio $5: 27$. Find the angle between the generator and the base plane. | ## Solution.
Let $A B C$ be the axial section of the given cone,
$O$ - the center of the sphere inscribed in it with radius $R, M$ and $N$ - the points of tangency of the sphere with the g... | \frac{\pi}{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,181 |
12.430 The surface of the sphere inscribed in a regular truncated triangular pyramid relates to the total surface area of the pyramid as $\pi: 6 \sqrt{3}$. Find the angle between the lateral face and the base plane of the pyramid. | Solution. Let $P_{1}$ and $P_{2}$ be the centers of the bases of the given regular truncated pyramid $A_{1} B_{1} C_{1} A_{2} B_{2} C_{2}, \quad O$ be the center of the sphere with radius $R$ inscribed in the pyramid, and $D_{1}$ and $D_{2}$ be the midpoints of the edges $B_{1} C_{1}$ and $B_{2} C_{2}$. Denote the side... | \operatorname{arctg}2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,182 |
12.431 The angle between the planes of two equal right triangles $A B C$ and $A D C$ with a common hypotenuse $A C$ is $\alpha$. The angle between the equal legs $\boldsymbol{A} \boldsymbol{B}$ and $A D$ is $\boldsymbol{\beta}$. Find the angle between the legs $B C$ and $C D$. | ## Solution.
Drop perpendiculars from vertex $B$ and vertex $D$ to $AC$: they will intersect at one point $M$, since $\triangle ABC = \triangle ADC$. Let $K$ be the midpoint of $BD$.
Sinc... | \angleBCD=2\arcsin\sqrt{\sin\frac{\alpha+\beta}{2}\cdot\sin\frac{\alpha-\beta}{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,183 |
12.432 The side of the lower base of a regular truncated quadrilateral pyramid is 5 times the side of the upper base. The lateral surface area of the pyramid is equal to the square of its height. Find the angle between the lateral edge of the pyramid and the plane of the base. | Solution.
Let $O_{1}$ and $O_{2}$ be the centers of the bases of the given regular truncated pyramid $A_{1} B_{1} C_{1} D_{1} A_{2} B_{2} C_{2} D_{2}$, with the sides of the bases being $2x$ and $10x$, and $K_{1}$ and $K_{2}$ be the midpoints of the edges $A_{1} D_{1}$ and $A_{2} D_{2}$. Then $K_{1} K_{2}$ is the heig... | \operatorname{arctg}\sqrt{9+3\sqrt{10}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,184 |
12.433 The base of a right prism is an isosceles trapezoid, the diagonals of which are perpendicular to the corresponding lateral sides. The angle between the diagonals of the trapezoid, opposite to its lateral side, is $\alpha$. The segment of a line connecting the vertex of the upper base with the center of the circl... | Solution.
Let the isosceles trapezoid \(ABCD\) be the base of the given prism, \(AB = CD\), \(O\) be the center of the circle circumscribed around \(ABCD\), and \(AC \perp CD\). Then \(AD\) ... | ^3\cos\beta\cdot\cos^2\frac{\alpha}{2}\cdot\sin2\beta\cdot\sin\alpha | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,185 |
12.434 The base of a right prism is a parallelogram with an acute angle $\alpha$. The diagonals of the prism form angles $\boldsymbol{\beta}$ and $\boldsymbol{\gamma}(\boldsymbol{\beta}<\boldsymbol{\gamma})$ with the base plane. Find the volume of the prism if its height is $\boldsymbol{H}$. | Solution.
Let the parallelogram $ABCD$ be the base of the given right prism, $\alpha=\angle A$. Then $AC > BD$, $AC = H \cdot \operatorname{ctg} \beta$, $BD = H \cdot \operatorname{ctg} \gamma \quad (AC$ and $BD$ are the projections of the diagonals of the prism). By the cosine theorem for $\triangle ABD$ and $\triang... | H^{3}\cdot\frac{\operatorname{tg}\alpha\cdot\sin(\gamma-\beta)\cdot\sin(\gamma+\beta)}{4\sin^{2}\beta\cdot\sin^{2}\gamma} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,186 |
12.435 The base of the prism is an equilateral triangle with side $a$. The lateral edge is equal to $b$ and forms angles $\alpha$ and $\beta$ with the intersecting sides of the base. Find the volume of the prism. | ## Solution.
Let $A_{1} D$ be the height of the given prism $A B C A_{1} B_{1} C_{1}$, where $\triangle A B C$ is equilateral, $\angle A_{1} A B = \alpha$, and $\angle A_{1} A C = \beta$. D... | \frac{^{2}b}{4}\sqrt{3-4(\cos^{2}\alpha-\cos\alpha\cdot\cos\beta+\cos^{2}\beta)} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,187 |
12.436 The base of the prism is a parallelogram with an acute angle $\alpha$. The lateral edge passing through the vertex of this angle $\alpha$ is equal to $b$ and forms angles, each equal to $\boldsymbol{\beta}$, with the adjacent sides of the base. Find the height of the prism. | ## Solution.
Let $A B C D A_{1} B_{1} C_{1} D_{1}$ be the given prism, $A A_{1}=b, \angle B A D=\alpha, \angle A_{1} A B=\angle A_{1} A D=\beta, \quad A_{1} O$ be the height of the prism.
... | \frac{b}{\cos\frac{\alpha}{2}}\sqrt{\sin(\beta+\frac{\alpha}{2})\cdot\sin(\beta-\frac{\alpha}{2})} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,188 |
12.437 The base of a right parallelepiped is a parallelogram with diagonals $a$ and $b(a>b)$ and an acute angle $\alpha$ between them. The smallest diagonal of the parallelepiped forms an acute angle $\beta$ with the larger diagonal of the base. Find the volume of the parallelepiped. | ## Solution.
Let \( A B C D A_{1} B_{1} C_{1} D_{1} \) be the given right parallelepiped and \( A C_{1} \) be its smaller diagonal, \( A C = b, \quad B D = a \). In the plane of the base \(... | V_{T}=\frac{^{2}\sin\alpha}{2\cos\beta}\cdot\sqrt{\sin(\beta+\alpha)\cdot\sin(\beta-\alpha)} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,189 |
12.438 The side of the base of a regular triangular pyramid is equal to $a$, and the dihedral angle at the base is $\alpha$. A right triangular prism is inscribed in this pyramid; three of its vertices lie on the apothems of the pyramid, and the other three - in the plane of the pyramid's base. Find the volume of the p... | ## Solution.
$C$
Let $O$ be the center of the sphere inscribed in the given regular pyramid $FABC$, $FP$ be its height, $FE$ and $FD$ be its apothems. Let $M$ and $N$ be two vertices of th... | \frac{^{3}\cdot\sin\frac{\alpha}{2}}{128\cdot\cos^{5}\frac{\alpha}{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,190 |
12.439 The base of a right prism is a rhombus. One of the diagonals of the prism is equal to a and forms an angle $\alpha$ with the base plane, and an angle $\beta$ with one of the lateral faces. Find the volume of the prism. | ## Solution.
A
Let $\quad T=A B C D A_{1} B_{1} C_{1} D_{1}$
- Given that the prism is a right prism, $B D_{1}=a$, and the base $A B C D$ is a rhombus, then $B B_{1}=a \cdot \sin \alpha, ... | V_{T}=\frac{^{3}\cdot\sin2\alpha\cdot\cos\alpha\cdot\sin\beta}{4\sqrt{\cos(\alpha+\beta)\cdot\cos(\alpha-\beta)}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,191 |
12.440 The ratio of two segments enclosed between parallel planes is $\boldsymbol{k}$, and the angles that each of these segments makes with one of the planes are in the ratio $2: 3$. Find these angles and the permissible values of $\boldsymbol{k}$. | Solution. Let the given segments be equal to $a$ and $k a$, and the angles they form with one of the planes be $x$ and $\frac{2}{3} x$ respectively. Since the planes are parallel, the distance between them is $a \cdot \sin x$ or $k a \cdot \sin \frac{2}{3} x$, i.e., $k \sin \frac{2}{3} x = \sin x$.
From this, $2 k \cd... | 2\arccos\frac{k+\sqrt{k^{2}+4}}{4};3\arccos\frac{k+\sqrt{k^{2}+4}}{4};\frac{2\sqrt{3}}{3}\leqk<\frac{3}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,192 |
12.441 The angle between the plane of square $A B C D(A B \| C D)$ and a certain plane $P$ is $\alpha$, and the angle between side $A B$ and the same plane is $\beta$. Find the angle between side $A D$ and plane $P$. | Solution. Without loss of generality, we can assume that the given plane $P$ passes through the vertex $A$ of the given square $ABCD$. Drop perpendiculars $BM$ and $DN$ to the plane $P$.
Le... | \arcsin\sqrt{\sin(\alpha-\beta)\cdot\sin(\alpha+\beta)} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,193 |
### 12.442 In a regular quadrilateral prism $A B C D A_{1} B_{1} C_{1} D_{1}$
$\left(A A_{1}\left\|B B_{1}\right\| C C_{1} \| D D_{1}\right)$ a plane is drawn through the midpoints of two adjacent sides of the base $D C$ and $A D$ and the vertex $B_{1}$ of the upper base. Find the angle between this plane and the pla... | ## Solution.
Let $B_{1} F N M E$ be the section of the given regular prism by a plane passing through $B_{1}$ and the midpoints $M$ and $N$ of the edges $A D$ and $D C$. Let point $K$ be t... | \arccos\frac{3}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,194 |
12.443 The distances from the center of the base of a regular quadrilateral pyramid to a lateral face and to a lateral edge are equal to $a$ and $\boldsymbol{b}$, respectively. Find the dihedral angle at the base of the pyramid. | ## Solution.
Let $O$ be the center of the given regular pyramid $F A B C D$. If $K$ is the midpoint of $C D$, then $C D \perp F K O$ (Theorem 11) $\Rightarrow F K O \perp F C D$ (Theorem 15)... | \arccos\frac{\sqrt{2^{2}-b^{2}}}{b} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,195 |
12.444 The base of the pyramid is an equilateral triangle. One of the lateral faces of the pyramid is perpendicular to the plane of the base. Find the cosine of the angle between the other two lateral faces, if they form an angle \(\alpha\) with the plane of the base. | Solution. Let $FABC$ be the given pyramid, $FAB \perp ABC$. Then the height $FK$ of the pyramid lies in the plane $FAB$. Let $FM$ be the height of $\triangle FBC$. Then $BC \perp KM$ (Theorem 12) $\Rightarrow$
Let $A B C A_{1} B_{1} C_{1}$ be the given prism. $A B B_{1} A_{1} \perp A B C, \angle B C A=90^{\circ}, \angle C B A=\alpha$. Let $B_{1} L$ and $A_{1} K$ be the heights of th... | \angleA_{1}MK=\operatorname{arctg}(\operatorname{tg}\alpha\cdot\operatorname{tg}\beta) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,197 |
12.446 Side $BC$ of triangle $ABC$, lying at the base of the inclined prism $ABC A_1 B_1 C_1 \left(A A_1 \left\| B B_1 \right\| C C_1\right)$, is equal to $a$, and the angles adjacent to it are $\beta$ and $\gamma$. Find the angle between the lateral edge and the plane of the base, if the volume of the prism is $V$ and... | ## Solution.
Let $A_{1} K$ be the height of the given prism $A B C A_{1} B_{1} C_{1}, \angle B=\beta, \angle C=\gamma$.
From the equality $A A_{1}=A_{1} B=A_{1} C$ it follows that $K$ is th... | \operatorname{arctg}\frac{4V\cdot\sin^{2}(\beta+\gamma)}{^{3}\cdot\sin\beta\cdot\sin\gamma} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,198 |
12.447 In a regular truncated triangular pyramid, two spheres are inscribed; one touches all its faces, the other touches all its edges. Find the sine of the angle between the lateral edge and the plane of the base. | ## Solution.
Let $A B C A_{1} B_{1} C_{1}$ be the given regular truncated pyramid, $P$ and $P_{1}$ be the centers of its bases $A B C$ and $A_{1} B_{1} C_{1}$, $b$ and $a$ be the sides of t... | \frac{1}{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,199 |
12.448 At the base of a quadrilateral pyramid lies an isosceles trapezoid with bases $a$ and $b(a>2 b)$ and an angle $\varphi$ between the equal segments of its diagonals. The vertex of the pyramid is projected onto the point of intersection of the diagonals of the base. The angles that the lateral faces passing throug... | ## Solution.
Since $P N = F P \cdot \operatorname{ctg} x$ and $P M = F P \cdot \operatorname{ctg} 2 x$, then
$\frac{P M}{P N} = \frac{b}{a} = (\Delta B C P \sim \triangle A D P) = \frac{\... | \frac{(b)^2}{24}\cdot\operatorname{tg}^2\frac{\varphi}{2}\cdot\sqrt{(2b)} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,200 |
12.449 The side of the base of a regular triangular pyramid is equal to $a$. The lateral face forms an angle $\alpha$ with the base plane. Find the distance between a lateral edge and a non-intersecting side of the base. | Solution.
Let $FO$ be the height of the given regular pyramid $FABC$, and $K$ be the midpoint of $AC$. Then $AC \perp FK, AC \perp BK \Rightarrow AC \perp FBK$ (Theorem 11) $\Rightarrow$ $AC ... | \frac{\sqrt{3}\cdot\tan\alpha}{2\sqrt{4+\tan^2\alpha}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,201 |
12.450 In a triangular pyramid, all faces are equilateral triangles. A plane is drawn through the side of the base, dividing the volume of the pyramid in the ratio 1:3, counting from the base. Find the angle between this plane and the plane of the base. | ## Solution.
Let the section $B C D$ of the given pyramid $F A B C$ divide its volume in the ratio 1:3
(counting from the base). Pyramids $FBCD$ and $ABCD$ have the same base $BCD \Rightarr... | \operatorname{arctg}\frac{\sqrt{2}}{5} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,202 |
12.451 In a regular quadrilateral pyramid, a plane is drawn through two lateral edges that do not belong to the same face. The ratio of the area of the section to the lateral surface area of the pyramid is $k$. Find the angle between two adjacent lateral faces and the permissible values of $\boldsymbol{k}$.
In a regul... | ## Solution.
Let $O$ be the center of the base of the given regular pyramid $FABCD$; $FAC$ is the section of the pyramid by a plane passing through $FA$ and $FC$. Drop perpendiculars from v... | \arccos(8k^2-1),0<k<\frac{\sqrt{2}}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,203 |
12.452 At the base of a right prism lies a parallelogram with an acute angle $\varphi$ between the diagonals. The diagonals of each of the adjacent lateral faces intersect at angles $\alpha$ and $\beta(\alpha>\beta)$, facing the corresponding sides of the base. Find the volume of the prism, if its height is $\boldsymbo... | Solution.
The lateral face $A A_{1} B_{1} B$ of the given right prism $T=A B C D A_{1} B_{1} C_{1} D_{1} (A B C D$ - its base) is a rectangle, in which $\angle B_{1} O B=\pi-\alpha$ and $B_{... | \frac{^{3}}{2}\cdot\operatorname{tg}\varphi\cdot\frac{\sin\frac{\alpha-\beta}{2}\cdot\sin\frac{\alpha+\beta}{2}}{\cos^{2}\frac{\alpha}{2}\cdot\cos^{2}\frac{\beta}{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,204 |
12.454 The side of the base of a regular quadrilateral pyramid is equal to $a$. The angle between the height of the pyramid and a lateral edge is $\alpha$ $\left(\alpha \leq \operatorname{arctg} \frac{\sqrt{2}}{2}\right)$. Find the area of the section of the pyramid by a plane passing through the midpoint of the height... | ## Solution.
Let $P$ be the midpoint of the height $FO$ of the given regular pyramid FABCD, and let KNLM be the section of the pyramid by a plane passing through $P$ and perpendicular to th... | \frac{^2}{8}\cdot\cos\alpha\cdot\operatorname{ctg}\alpha\cdot\operatorname{tg}2\alpha | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,205 |
12.455 Let $A B$ be the diameter of the lower base of the cylinder, $A_{1} B_{1}$ a chord of the upper base, parallel to $\boldsymbol{A B}$. The plane passing through the lines $A B$ and $A_{1} B_{1}$ forms an acute angle $\alpha$ with the plane of the lower base of the cylinder, and the line $A B_{1}$ forms an angle $... | ## Solution.
Let $D, C, P$ be the projections on the lower base of the given cylinder of points $B_{1}, A_{1}, O_{1}$, respectively, where $O_{1}$ is the midpoint of the chord $A_{1} B_{1}$.... | 2R\operatorname{tg}\beta\cdot\frac{\sqrt{\sin(\alpha+\beta)\cdot\sin(\alpha-\beta)}}{\sin\alpha\cdot\cos\beta} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,206 |
12.465 The height of a regular triangular pyramid SABC is $\boldsymbol{H}$. Through the vertex $A$ of the base $ABC$, a plane is drawn non-perpendicular to the opposite lateral edge SC. This plane forms an angle $\alpha$ with the base plane. Find the volume of the part of the pyramid enclosed between the base plane and... | ## Solution.
Let SO be the height of the given regular pyramid $SABC$. Drop perpendiculars from vertices $A$ and $B$ to the edge $SC$: they will intersect at one point $D$ (since the later... | \frac{3\sqrt{3}}{8}H^3\tan^2\alpha\cdot\sin^2\alpha | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,207 |
12.457 The height of a regular truncated square pyramid is $\boldsymbol{H}$. The lateral edge forms an angle $\alpha$ with the base, and the diagonal of the pyramid forms an angle $\beta$ with the base. Find the area of the section of the pyramid by a plane passing through the diagonal of the pyramid parallel to the no... | ## Solution.
Let $A B C D A_{1} B_{1} C_{1} D_{1}$ be the given regular truncated pyramid, and $O$ be the projection of $B_{1}$ onto the lower base $A B C D$. Then $\angle B_{1} D O=\beta, ... | S_{B_{1}LDK}=\frac{H^{2}\sin(\alpha+\beta)\cdot\sin(\alpha-\beta)}{\sin^{2}\alpha\cdot\sin\beta\cdot\sin2\beta} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,208 |
### 12.458 The sides of the lower and upper bases of a regular triangular truncated pyramid are equal to $a$ and $\boldsymbol{b}(a>b)$, respectively. The lateral face forms an angle $\alpha$ with the plane of the base. Find the area of the section of the pyramid by a plane passing through the midline of the lateral fac... | ## Solution.
Let $MN$ be the midline of the lateral face of the given regular truncated pyramid $ABC A_1 B_1 C_1$, $O$ and $O_1$ be the centers of the bases of the pyramid, and $MN L K$ be t... | S_{MNLK}=\frac{7a+3b}{144\cos\alpha}\cdot\sqrt{3(^2+b^2+2ab\cos2\alpha)} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,209 |
12.459 Find the radius of the sphere inscribed in a regular triangular pyramid, the height of which is $H$, and the angle between the lateral edge and the base plane is $\alpha$. | ## Solution.
Let $O$ be the center of the sphere of radius $x$ inscribed in the given regular pyramid $F A B C$, $F K$ the height of the pyramid, and $D$ the midpoint of $B C$. Then $B C \p... | \frac{H}{4\operatorname{tg}^{2}\alpha}(\sqrt{4\operatorname{tg}^{2}\alpha+1}-1) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,210 |
12.460 The radius of the sphere circumscribed around a regular quadrilateral pyramid is to the side of the base as 3:4. Find the angle between the lateral face and the base plane. | Solution.
Let $O$ be the center of the sphere circumscribed around the given regular pyramid $F A B C D$, $F K$ be the height of the pyramid, $a = A B$, and $P$ be the midpoint of $C D$. A... | \operatorname{arctg}2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,211 |
12.461 A cylinder is inscribed in a cone, the axial section of which is a right triangle; its lower base lies in the plane of the cone's axis. The ratio of the lateral surfaces of the cone and the cylinder is $4 \sqrt{2}$. Find the angle between the plane of the base of the cone and the line passing through the center ... | Solution.
Let $\triangle A O B$ be the axial section of the given cone $\left(\angle A O B=90^{\circ}\right)$, and $M N K L$ be the axial section of the cylinder inscribed in it. Let $O P$ be... | \operatorname{arcctg}(4\2\sqrt{2}) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,212 |
12.462 The base of the pyramid is an isosceles trapezoid with an acute angle $\alpha$. This trapezoid is circumscribed around the base circle of a cone. The vertex of the pyramid lies on one of the generatrices of the cone, and its projection onto the base plane coincides with the point of intersection of the diagonals... | Solution.
Let the isosceles trapezoid $ABCD$ $(BC \| AD, BC < AD)$ be the base of the given pyramid $FABCD$, $O$ be the center of the base of the given cone, $K$ be the vertex of the cone, ... | \frac{2}{3}^3\operatorname{tg}\frac{\alpha}{2}\cdot\sin\beta\cdot\sin2\beta | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,213 |
12.463 The base of the pyramid FABC is an isosceles triangle $ABC$, where the angle between the equal sides $AB$ and $AC$ is $\alpha (\alpha < \pi / 2)$. A triangular prism $AED A_{1} E_{1} D_{1}$ is inscribed in the pyramid; points $A_{1}, E_{1}$, and $D_{1}$ lie on the lateral edges $AF, CF$, and $BF$ of the pyramid,... | Solution.
By Theorem 14, $B C\|D E\| D_{1} E_{1}$. Let $h_{1}$ and $h_{2}$ be the heights of the prism $T_{1}=A E D A_{1} E_{1} D_{1}$ and the pyramid $T_{2}=F A B C$. From the similarity o... | \frac{\cos\alpha}{8\cos^{6}\frac{\alpha}{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,214 |
12.465 The ratio of the volume of a regular truncated triangular pyramid to the volume of a sphere inscribed in it is $k$. Find the angle between the lateral face of the pyramid and the plane of the base and the permissible values of $k$. | ## Solution.
Let $P$ and $P_1$ be the centers of the bases of the given regular truncated pyramid, $M M_1$ be the apothem of one of its faces, $O$ be the center of the sphere inscribed in t... | \operatorname{arcctg}\frac{\sqrt{2\sqrt{3}\pik-27}}{6}, | Geometry | math-word-problem | Yes | Yes | olympiads | false | 52,216 |
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