📐 geometry

1. **Stating the problem:** We have a regular hexagon with side length $16$ and an inscribed right triangle sharing one side of the hexagon as its base. The triangle's height is $1

1. **State the problem:** Find the distance between the points $(-10.6, -17.8)$ and $(15.4, 8.1)$. 2. **Formula used:** The distance $d$ between two points $(x_1, y_1)$ and $(x_2,

1. **State the problem:** Find the distance between the points $ (16, -8) $ and $ (-11, -20) $ on the coordinate plane. 2. **Formula used:** The distance $ d $ between two points $

1. **State the problem:** Given two parallel lines $p$ and $q$ cut by a transversal, we are to understand the relationships between angles $\angle 1$, $\angle 2$, and $\angle 3$. 2

1. **Problem Statement:** Given two parallel lines $p$ and $q$ cut by a transversal $w$, with $m\angle 1 = 55^\circ$, find the measures of $\angle 2$ and $\angle 3$ and explain the

1. **State the problem:** We have a regular polygon with 12 sides (a dodecagon). One interior angle is given as $23x - 11$ degrees. 2. **Find the sum of the interior angles:** The

1. **Problem statement:** We have a right triangle with points G, H, and F, where GH is parallel to DE. Given that GH is 7 more than EH, FH = 6, and DE = 15, we need to find the le

1. **Stating the problem:** Determine whether the given pairs of triangles are similar or not, and if similar, identify the similarity criteria. 2. **Similarity criteria for triang

1. The problem states that two vertical angles are formed by intersecting lines, with one angle measuring 145° and the opposite angle measuring (2x + 15)°. 2. Vertical angles are a

1. **Stating the problem:** We are given two angles formed by a diagonal line crossing two parallel horizontal lines. The angles are labeled as $(15x - 2)^\circ$ and $(11x + 34)^\c

1. **State the problem:** We are given two angles formed by a transversal intersecting two parallel lines. The angles are labeled as $5x + 6$ degrees and $8x - 39$ degrees. 2. **Id

1. **Problem:** Select all the sets of conditions that will form only one unique triangle. 2. **Understanding the problem:** We need to identify which sets of given conditions guar

1. **State the problem:** We are given two triangles, Triangle A with sides 10, 18, and 15, and Triangle B with sides 6, 4, and 7.2. We need to find the scale factor from figure A

1. **State the problem:** We need to find the total cost to pave all the grey areas in the picnic area. Each grey shape's area must be calculated, then summed, and finally multipli

1. **State the problem:** We are given the area of a square as 36 cm² and need to find the length of one side of the square. 2. **Formula used:** The area $A$ of a square is given

1. **State the problem:** We have a ladder leaning against a building forming a right triangle. The ladder is the hypotenuse of length 35 feet, the building height is one leg of le

1. **State the problem:** We need to find the area of a right trapezoid with bases 12.5 ft and 6 ft, and height 6 ft. 2. **Formula for the area of a trapezoid:**

1. **State the problem:** We need to find the area of an L-shaped composite figure with outer dimensions 14 inches by 14 inches and an inner cutout of 7 inches by 7 inches. 2. **Fo

1. The problem asks to write three proportions involving the geometric mean from the similar right triangles $\triangle GIF$ and $\triangle GHF$. 2. In right triangle similarity wi

1. **State the problem:** We have a right triangle with one leg of length 30, hypotenuse 40, and the other leg divided into two segments: one segment is $x$ (vertical segment), and

1. **State the problem:** We are given two triangles, one larger and one smaller, where the larger triangle is a dilation of the smaller triangle with scale factor $k=2.5$. We need