📐 geometry

1. Задачата е да определим ъглите между правите в правоъгълен паралелепипед ABCDA1B1C1D1 с дадени размери $BC=2$ и $AA1=4$. Точка $P$ е на ръба $DD1$ с $DP=1$, точка $M$ е среда на

1. **State the problem:** We have two similar polygons, Polygon A and Polygon B.

1. **State the problem:** We have triangle \(\triangle ABC\) with vertices \(A(-2, 2)\), \(B(-8, 7)\), and \(C(-7, -2)\). We apply a sequence of transformations: rotate 90° about t

1. O problema pede para calcular o comprimento da diagonal BD do retângulo ABCD, onde os pontos A = (0,0), B = (1,4) e C = (3,6) são vértices consecutivos. 2. Primeiro, lembre-se q

1. **Stating the problem:** We have a right triangle $\triangle ABC$ with $m(\angle B) = 90^\circ$, $m(\angle C) = 60^\circ$, and side $BC = 4$ cm.

1. **Stating the problem:** We have a triangle with side lengths 12.5 cm, 10 cm, and 8.5 cm. A similar triangle has its longest side equal to 20 cm. We need to find the ratio of th

1. The problem involves understanding the shape formed by sides that are quarter circles with radius 5 cm. 2. A quarter circle is a sector of a circle with a 90-degree angle, and i

1. **Problem statement:** We have a rectangle ABCD with side lengths 10 cm by 10 cm. Two quarter circles of radius 5 cm are cut out from opposite corners, leaving a shaded figure w

1. **State the problem:** We have a rectangular box with dimensions 8 inches, 20 inches, and 40 inches. The dimensions of the box are reduced by half, and we need to find the const

1. **State the problem:** Find the surface area of the L-shaped solid composed of two rectangular prisms joined together. 2. **Identify dimensions:**

1. **State the problem:** Find the surface area of a composite solid made of two rectangular prisms joined together. 2. **Identify dimensions:**

1. **State the problem:** We are given volumes of spheres A and B, and the ratio of the radii of spheres B and C. We need to find the ratio of the surface areas of spheres A and C.

1. The problem asks to find the area, but the shape or figure is not specified. 2. To find the area, we need to know the type of figure (e.g., rectangle, triangle, circle) and its

1. **Stating the problem:** We have a shape composed of 9 connected 1 cm by 1 cm squares arranged in a step-like pattern on a grid. We want to find the area of this shape. 2. **For

1. **State the problem:** Find the volume of a beach ball with a diameter of 2 ft, using $\pi = 3.14$. 2. **Formula:** The volume $V$ of a sphere is given by

1. **State the problem:** Find the volume of a cylinder with radius $r=2$ cm and height $h=6$ cm. 2. **Formula for the volume of a cylinder:**

1. **State the problem:** We need to solve for the value of $q$ given two angles: $(q + 2)^\circ$ and $(3q - 2)^\circ$. 2. **Analyze the angles:** The problem suggests these two an

1. **State the problem:** Calculate the total area of the compound figure composed of a red triangle, a blue rectangle, two yellow right triangles, and a small trapezoid as describ

1. **Problem Statement:** We have triangle $\triangle ABC$ and its dilation $\triangle A'B'C'$ about point $P$ with scale factor 2. We need to determine if the following claims are

1. **Problem Statement:** We are given two similar triangles and need to identify which triangles are similar and then solve for the missing side lengths. 2. **Formula and Rules:**

1. **Problem Statement:** We have two triangles with vertices J, N, K and K, L, M respectively. Given sides are $JN=15$, $NK=5$, $KL=9$, and $LM=10$. We need to find the missing si