📐 geometry

1. **State the problem:** We have an isosceles triangle $PQR$ with base $RQ = 18$ cm and area $108$ cm$^2$. We need to find the length of side $PQ$.

1. **State the problem:** We need to find the area of a composite shape made of rectangle ABDE and right-angled triangle BCD. 2. **Identify given dimensions:**

1. **State the problem:** We have a ladder leaning against a tree forming a right-angled triangle. The ladder is the hypotenuse with length $15$ m, the height up the tree is $14$ m

1. **State the problem:** We have a right-angled triangle with legs of lengths $15$ cm and $20$ cm, and hypotenuse $x$. We need to find $x$ and then calculate the perimeter. 2. **F

1. **State the problem:** We need to find the length of the hypotenuse $QR$ in right triangle $PQR$ where the right angle is at $P$. The legs are $PQ = 7$ cm and $PR = 3$ cm. 2. **

1. **State the problem:** A ladder of length $3$ m (which is $300$ cm) is resting against a vertical wall. Initially, the foot of the ladder is $70$ cm from the wall. The ladder sl

1. **Problem 1:** Given a right-angled triangle with one leg $3cm$ and hypotenuse $10\sqrt{2}cm$, find the other leg. 2. **Formula:** Use the Pythagorean theorem: $$c^2 = a^2 + b^2

1. **Problem 1: Identify which angle statements are true given the diagram with lines AB, DE, FC intersecting at G and \(\angle ACE\) a right angle.\n\n2. **Key definitions:**\n- V

1. **Problem Statement:** We are given a rectangle $ABCD$ with vertices $A$ (top-left), $B$ (top-right), $C$ (bottom-right), and $D$ (bottom-left). Both diagonals $AC$ and $BD$ are

1. **State the problem:** We are given that the areas of two similar plane figures are in the ratio $$36:25$$. We need to find the ratio of their perimeters. 2. **Recall the proper

1. Let's clarify the problem you want to solve involving circles. 2. Typically, problems with circles involve formulas such as the circumference $C = 2\pi r$ or the area $A = \pi r

1. **Problem Statement:** We have a square whose two adjacent sides are divided into $n$ equal parts. Lines are drawn connecting these division points on one side to the division p

1. **Problem:** Determine the area of the shaded portion of the circle. 2. **Formula used:** For a circular sector, the area is

1. **State the problem:** We need to find which point on the line segment $JW$ is closest to the point $G(5,3)$. The points given are $J(0,0)$, $W(-5,-3)$, and $G(5,3)$. The option

1. **State the problem:** We are given two lines, $a$ and $g$, intersecting a transversal $w$ at two points, creating angles labeled $11, 7, 8, 1$ at the upper intersection and $9,

1. **State the problem:** We are given two right triangles $\triangle ABC$ and $\triangle PQR$ with $\angle B$ and $\angle Q$ right angles, $\angle A \cong \angle P$, and side $AB

1. **State the problem:** We need to find the length of segment $PQ$ given that point $Q$ lies on segment $PR$ with $PQ = 3x - 9$, $PR = 2x$, and $QR = 5$. 2. **Recall the segment

1. **State the problem:** We need to find the area of a triangle with side lengths $20$ cm, $21$ cm, and $29$ cm, where the right angle is at the vertex between the sides of length

1. **State the problem:** We are given a rectangle divided by its diagonal into two triangles. The shaded triangle has an area of $42\ \text{cm}^2$. We need to find the area of the

1. **State the problem:** We need to prove that $m\angle PRS + m\angle VUW = 180^\circ$ given the parallel lines and angle relationships.

1. **State the problem:** We need to prove that $$m\angle PRS + m\angle VUW = 180^\circ$$ given the parallel lines and angle relationships.