📐 geometry

1. **State the problem:** Translate the triangle with vertices A(2,4), B(1,2), C(4,2) by the vector (-4,1).

1. **State the problem:** We have two triangles, each with angles labeled. We want to find the measure of angle $z$. 2. **Recall the triangle angle sum rule:** The sum of the inter

1. **State the problem:** We need to find the measure of angle \(\angle K\) in triangle \(KLM\).\n\n2. **Recall the triangle angle sum rule:** The sum of the interior angles in any

1. **Stating the problem:** We are given two triangles, \(\triangle HOT\) and \(\triangle PIG\), which are similar (\(\triangle HOT \sim \triangle PIG\)). We want to find the relat

1. The problem states that triangles $\triangle SIT$ and $\triangle DOG$ are similar, denoted as $\triangle SIT \sim \triangle DOG$. 2. In similar triangles, corresponding angles a

1. **Problem Statement:** We have two similar triangles, ABC and XYZ. We know that \(\angle Y = 30^\circ\) in triangle XYZ, and we want to find the measure of \(\angle B\) in trian

1. **State the problem:** We have a right triangle with a hypotenuse of length 10 feet and one angle of 60°. 2. **Identify the sides:** In a right triangle with angles 30°, 60°, an

1. **Problem statement:** Find the missing side length, area, and perimeter of a right isosceles triangle with one leg length $\sqrt{2}$ meters and one angle of 45°.

1. **Problem:** Monica's TV screen is 44 inches wide and the diagonal measures 55 inches. Find the area of the TV screen. 2. **Formula:** For a rectangle, the diagonal $d$, width $

1. **State the problem:** We have a triangle with sides $TS=4$ cm, $TR=12.9$ cm, and angle $\angle TSR=77^\circ$. We need to find the other two angles of the triangle.

1. The problem is to find the angles of a triangle or geometric figure, but since no specific figure or values are given, let's assume a common problem: finding angles in a triangl

1. **State the problem:** Calculate the area of triangle STR where side ST = 4 cm, side TR = 12.9 cm, and the included angle \(\angle STR = 77^\circ\).

1. **State the problem:** Find the area of triangle PQR with sides $QR=8.7$ cm, $RP=8$ cm, and base $QP=6$ cm. 2. **Formula used:** We use Heron's formula for the area of a triangl

1. **State the problem:** We are given a triangle with sides $a=19$ cm, $c=21$ cm, and the segment $s_{156}=15$ cm (likely a side or segment related to the triangle). We want to so

1. **State the problem:** We have a toy made from a cone and a hemisphere. Both have radius $r$ cm. The slant height of the cone is $3r$ cm. The total surface area of the toy is 30

1. **State the problem:** Find the area of a triangle with sides $b=15$ cm, $c=24$ cm, and angle $\angle A=103^\circ$ between them. 2. **Formula used:** The area $A$ of a triangle

1. **State the problem:** We are given the midpoint $M(6.5, 3.5)$ of segment $\overline{RS}$ and one endpoint $S(15, -12)$. We need to find the coordinates of the other endpoint $R

1. **State the problem:** We are given the midpoint $M(-17, 6.5)$ of segment $\overline{JK}$ and one endpoint $K(-15, -6)$. We need to find the coordinates of the other endpoint $J

1. **State the problem:** We are given the midpoint $M(6.5, 11)$ of segment $\overline{KL}$ and one endpoint $L(13, 16)$. We need to find the coordinates of the other endpoint $K$.

1. **State the problem:** We are given the midpoint $M(2.5, -6.5)$ of segment $\overline{PQ}$ and one endpoint $P(-5, -15)$. We need to find the coordinates of the other endpoint $

1. **Problem 1: Find the length of KL** Given points I, J, K, L on a line with lengths: