📐 geometry

1. **State the problem:** Given a circle with radius $7$ cm and diameter $TC$, find the circumference, area, various angle measures, arc measures, arc lengths, and the area of the

1. **State the problem:** We have a right triangle $\triangle PQR$ with right angle at $R$. A segment $QS$ is drawn perpendicular to $PR$, intersecting at $S$. We need to complete

1. **State the problem:** We have a right triangle $\triangle PQR$ with a right angle at $R$. Point $S$ lies on segment $PQ$ such that two smaller right triangles are formed inside

1. **Problem Statement:** We are given a right triangle \(\triangle DFG\) with \(FE\) perpendicular to the hypotenuse \(DG\). We need to determine which triangles are similar to \(

1. **State the problem:** We have a right triangle with base $a$ and hypotenuse 18. A segment from the right angle perpendicular to the hypotenuse divides it into two parts, the la

1. **Problem 1: Which triangles are similar to \(\triangle DEF\)?**\n\nGiven the description, \(\triangle DEF\) is a right triangle inside a larger right triangle \(\triangle DFG\)

1. **Problem Statement:** Given triangle $\triangle PQR$ with point $S$ on side $PQ$ such that $PS$ is perpendicular to $SR$ and $RS$ is perpendicular to $QR$, complete the table o

1. **State the problem:** We have a right triangle with base $a$ and hypotenuse 18. A segment from the right angle perpendicular to the hypotenuse divides it into two parts, the la

1. **State the problem:** Find expressions for the total surface area and volume of a rectangular packing crate with dimensions:

1. **State the problem:** We need to find the difference in volume between two cubes built by Hector and Karin with side lengths 12 in. and 14 in. respectively.

1. **Problem statement:** From an external point $P$, tangents $PA$ and $PB$ are drawn to a circle with center $O$. Given that $\angle PAB = 50^\circ$, find $\angle AOB$. 2. **Key

1. **Problem Statement:** Find the area of a figure composed of a rectangle and one semicircle added on the right side, with another semicircle removed on the left side. The rectan

1. **State the problem:** Find the area of a figure composed of a rectangle and two semicircles attached to the shorter sides of the rectangle. 2. **Given dimensions:** Length of r

1. **Problem statement:** In parallelogram PQRS, given that diagonal SQ = 26, find the length of segment ST, where T is the intersection of diagonals PR and SQ. 2. **Key property:*

1. The problem states that triangles \(\triangle CJW \cong \triangle AGS\), with \(m\angle A = 50^\circ\), \(m\angle J = 45^\circ\), and \(m\angle S = 16x + 5\). We need to find \(

1. **Problem statement:** We are given a triangle \(\triangle ABC\) with sides \(AB = 7.5x\), \(BC = 6x + 3\), and \(AC = 10x - 5\). The triangle is equilateral, meaning all sides

1. **State the problem:** We have a right triangle ABC with vertices A at the origin $(0,0)$, C on the x-axis, and B on the hypotenuse line $y = -0.5x + k$. The left leg is $y = 2x

1. **State the problem:** We have two similar triangles ABC and EBD with sides AB = 9 ft, AC = h, DE = 7 ft, and BE = 15 ft. We need to find the unknown side length $h$.

1. **State the problem:** We have a right triangle PQR with sides PQ = 20, QR = 21, and PR = 13. The height $h$ is drawn from the top vertex P perpendicular to the base QR. We need

1. **Stating the problem:** We have a sequence of figures where each figure is a large triangle subdivided into smaller triangles. The first figure has 1 small triangle, the second

1. **State the problem:** We have a right triangle where one leg is 1 millimeter longer than the shorter leg, and the hypotenuse is 2 millimeters longer than the shorter leg. We ne