📐 geometry

1. **State the problem:** We need to prove that the line segment $\overline{HI}$ is congruent to the line segment $\overline{XW}$ by using the given triangles $WVX$ and $HJI$ and t

1. The problem asks: How many triangles are there in an octahedron? 2. An octahedron is a type of polyhedron with 8 triangular faces and 6 vertices.

1. The problem asks how to determine if pool ABCD is similar to pool EFGH using geometric transformations. 2. Similarity between two shapes means one can be obtained from the other

1. The problem states that triangle $\triangle ABC$ is dilated from point $A$ by a scale factor of $\frac{1}{2}$. We need to find the relationship between segment $D'E'$ (the image

1. The problem asks to draw the net of each figure and calculate its surface area. 2. For figure (a), a triangular pyramid with edges 8 cm, 6 cm, and 6 cm:

1. **State the problem:** We have a segment $IJ$ with length 2, and it is dilated by a scale factor of $\frac{3}{4}$ to form segment $I'J'$. We need to find the length of $I'J'$.\n

1. The problem states that the measure of angle $\angle HIJ$ is given as 94°. 2. It also states that $m \angle HIJ = 90^\circ$, which is a contradiction because an angle cannot be

1. The problem involves finding the missing side lengths in right triangles with given angles and side lengths, specifically using 30° and 60° angles. 2. For a 30°-60°-90° triangle

1. **Problem statement:** Given a right triangle ABC with a right angle at B, M is the midpoint of AC. From M, MH is drawn perpendicular to AB at H, and MK is drawn perpendicular t

1. The problem involves finding the value of $x$ given angles in a circle, where angles such as $x^\circ$, $2x^\circ$, $3x^\circ$, $-x^\circ$, and $130^\circ$ appear in various con

1. **Problem Statement:** Explain the properties of a quadrilateral, specifically focusing on the sum of interior angles and the diagonals forming triangles.

1. **Stating the problem:** We need to find the length of segment $JN$ by using the similarity of two right triangles $\triangle JKL$ and $\triangle JNM$. 2. **Identifying similar

1. **Problem Statement:** We need to find the length of segment $JN$ in the given right triangle configuration. 2. **Given:**

1. **State the problem:** We need to classify triangle $\triangle WXY$ with vertices $W(-7,1)$, $X(0,-4)$, and $Y(2,3)$ as equilateral, isosceles, scalene, or none of the above. 2.

1. **State the problem:** We are given a triangle with three angles labeled as $ (8x - 4)^\circ $, $ (17x - 23)^\circ $, and $ (3x + 17)^\circ $. We need to find the value of $x$.

1. Problem 6a: Find the composition of transformations $T_{(3,-2)} \circ T_{(1,-1)}$. 2. The rule for composition of translations $T_{(a,b)} \circ T_{(c,d)}$ is:

1. Problem: Write a reflection rule that maps each triangle to its image. 2. Reflection rule means finding the line or axis over which the triangle is flipped.

1. **Stating the problem:** Prove or verify the equation $$\frac{AB}{AE} + \frac{AD}{AF} = 1$$ where points and segments are part of a geometric figure involving parallelogram $ABC

1. **State the problem:** We have two hemispheres and a right prism. The total volume of the two hemispheres equals the volume of the prism. We need to find the value of $p$. 2. **

1. সমস্যাটি হলো: একটি গোলকের ব্যাসার্ধ $R$ এবং একটি লম্ববৃত্তাকার চোঙের ব্যাসার্ধ $r$ এবং উচ্চতা $\frac{9}{2}r$ দেওয়া আছে। গোলক এবং চোঙের আয়তন সমান হলে $R:r$ এর অনুপাত নির্ণয় করত

1. **Problem Statement:** Prove that points D, A, C, and E lie on the same circle, i.e., quadrilateral DACE is cyclic. 2. **Key Property:** A quadrilateral is cyclic if and only if