📐 geometry

1. **State the problem:** We have a triangle with sides labeled $r$, 17, and 5, and an angle of $32^\circ$ opposite side $r$. We need to find the length of side $r$. 2. **Identify

1. **Problem statement:** We are given a triangle PQR with sides $PQ=15$, $PR=9$, and the included angle $\angle P=136^\circ$. We need to find the length $p$ of the side opposite a

1. **Problem statement:** Reflect the given shapes in the specified lines.

1. The problem asks which statement must be true about triangles that are similar. 2. By definition, similar triangles have the same shape but not necessarily the same size.

1. The problem states that triangle ABC is similar to triangle FGH, and we need to find the proportion to calculate the length of segment BC in centimeters. 2. Similar triangles ha

1. **Problem Statement:** We are given two similar quadrilaterals PQRS and TUVW. We need to determine which statement about their angles or side ratios must be true based on simila

1. The problem states that two triangles are similar, meaning their corresponding sides are proportional. 2. We are given the larger triangle with sides 15, 24, and 27, and the sma

1. **State the problem:** We have a right triangle with two sides given: one side is 13 units, and the other perpendicular side is 12 units. We need to find the length of the third

1. **Problem statement:** We have a right triangle with one leg of length 8 and a hypotenuse of length $2\sqrt{41}$. We need to find the length of the other leg. 2. **Formula used:

1. The problem asks to determine if the ratio $12 : 12\sqrt{3} : 24$ corresponds to the side lengths of a $45^\circ$-$45^\circ$-$90^\circ$ triangle, a $30^\circ$-$60^\circ$-$90^\ci

1. The problem asks whether the statement "A set of coordinates that indicates the location of a point on a coordinate plane is a coordinate" is True or False. 2. Let's analyze the

1. **Stating the problem:** We are given a right triangle with one leg of length 6 and the hypotenuse of length $3\sqrt{13}$. We want to find the length of the other leg. 2. **Form

1. **State the problem:** We need to find the measure of angle $\angle E$ in the given right triangle. 2. **Identify the triangle sides:** The triangle has a right angle at $D$. Th

1. The problem is to find the measure of angle $\angle E$ in a right triangle with vertices $D$, $F$, and $E$, where $\angle D$ is the right angle. 2. Given: $DF = 3$ inches, $FE =

1. **State the problem:** We have a right triangle with a vertical leg of length 9.7 km, a hypotenuse of length 9.4 km, and a missing horizontal leg $b$. We need to find $b$. 2. **

1. **State the problem:** We have a right triangle with one leg measuring 8.7 cm, the hypotenuse measuring 8.9 cm, and the other leg labeled as $a$. We need to find the length of t

1. **State the problem:** We need to classify the given triangle based on its side lengths and angles. 2. **Analyze the triangle:** The triangle has one right angle (indicated by t

1. **State the problem:** We are given a triangle PQR with sides \(|PQ|=15\) cm, \(|PR|=12\) cm, and \(|RQ|=5\) cm. We need to find the measure of angle \(\angle PQR\) to the neare

1. The problem involves understanding and labeling the views of a 3-D stepped block object: the top, side, and front views. 2. The top view shows a 2x3 rectangle of cubes, represen

1. The problem asks to find the measure of \(\angle ABC\) given that \(BD\) is an angle bisector in triangle \(ABC\) with \(\angle ABD = x + 6\) and \(\angle DBC = 2x - 3\). 2. Sin

1. **Problem 1:** Decide if the altitudes of a triangle always intersect inside the triangle. The altitudes of a triangle are the perpendicular lines from each vertex to the opposi