📐 geometry

1. The problem states that triangles $\triangle GHI$ and $\triangle JKL$ are congruent, i.e., $\triangle GHI \cong \triangle JKL$. 2. By the definition of congruent triangles, corr

1. **State the problem:** We need to find the surface area of a square-based pyramid with a square base side length of 30 mm and four identical triangular faces each having a heigh

1. **State the problem:** We need to find the surface area of a square-based pyramid where the base edges are 6 cm and each triangular face has a height of 19 cm. 2. **Formula for

1. **State the problem:** We have a regular 9-sided polygon (nonagon) with interior angles of 140 degrees and exterior angles of 40 degrees. We want to find the angle $x$ formed at

1. **Problem statement:** We have a circle with radius $5$ cm and a chord of length $6$ cm. We need to find the area of another circle concentric with the first one that is tangent

1. **Problem statement:** Given a circle $(O)$ and a point $A$ outside it, two tangents $AB$ and $AC$ touch the circle at points $B$ and $C$ respectively. $H$ is the intersection o

1. **State the problem:** We have two concentric circles. The outer circle has a radius of 10 cm. The area of the inner circle is half the area of the outer circle. We need to find

1. **Problem statement:** We need to find the area of a swimming pool formed by two identical circles of radius 9 m, where each circle passes through the center of the other. 2. **

1. Problem statement: Consider parallelogram QRST with diagonals intersecting at U. 2. Given: the angle at vertex R formed by RU and RS is $32^\circ$.

1. **State the problem:** We have parallelogram VWXY with diagonals WY and VX intersecting at Z. Given segment WZ = 9, angle X = 64°, angle Y = 39°, and segment ZY = 4x + 1, find $

1. **State the problem:** We have a triangle with side lengths 13, 16, and $x$. We need to find the possible values of $x$ such that these lengths can form a triangle. 2. **Recall

1. **State the problem:** We have a triangle with side lengths 10, 14, and $x$. We need to find the possible values of $x$ such that these lengths can form a triangle. 2. **Recall

1. **State the problem:** We have a triangle with side lengths 11, 12, and $x$. We need to find the possible values of $x$ such that these lengths can form a triangle. 2. **Recall

1. **Problem statement:** We have a circle with radius $2\sqrt{2}$ and a shaded right triangle formed by two radii and a chord subtending a $135^\circ$ angle at the center. 2. **Go

1. The problem states: In parallelogram ABCD, point E lies on CD, and we need to find angle \(\angle ABC\).\n\n2. Recall that in a parallelogram, opposite angles are equal and adja

1. **State the problem:** Find the distance between points $M(-9, -6)$ and $N(-9, -10)$ on the coordinate plane. 2. **Formula used:** The distance between two points $(x_1, y_1)$ a

1. **Problem statement:** If the side length of a square is increased four times, find the percent increase in the area of the square. 2. **Formula for area of a square:** The area

1. The problem asks about the solid formed when a parallelogram is revolved completely around a given straight line. 2. When a 2D shape is revolved around a line, the 3D solid form

1. نبدأ بقراءة المسألة: لدينا معادلتان من شرطين لعمود منصفين: $$PQ = 2x = 9$$

1. **بيان المسألة:** نحن ندرس خصائص مثلث له نقاط X, Y, W, Z حيث W تقع على القطعة XZ.

1. **State the problem:** We have a right circular cylinder with radius $r$ and height $h$. A second cylinder has volume 392 times the volume of the first. We want to find possible