📐 geometry

1. The problem states that the area of triangle $ADE$ equals the area of quadrilateral $DEBC$, and both are half the area of triangle $ABC$. It is given that this half area equals

1. **Problem statement:** We have a sector ABC of a circle with radius $a$ and central angle $\angle BAC = \frac{\pi}{6}$. Points D and E lie on AB and AC such that $AD = AE = ka$

1. The problem involves understanding a 3D block pyramid made of 1 cm x 1 cm x 1 cm cubes, where the base layer has 9 cubes in a row, and each upper layer decreases by one cube unt

1. **Problem statement:** Jon is tiling a 2-foot wide border around a bathtub that measures 5 feet by 3 feet. Each tile measures 4 inches by 4 inches. We need to find the minimum n

1. **Problem statement:** Jon is tiling a 2-foot wide border around a bathtub that measures 5 feet by 3 feet. Each tile measures 4 inches by 4 inches. We need to find the minimum n

1. **State the problem:** We need to find the number of sides $n$ of a regular polygon whose interior angle is $150^\circ$. 2. **Formula used:** The measure of each interior angle

1. **Complementary Angles** A complementary angle pair sums to 90°.

1. **Problem Statement:** We have a kite-shaped quadrilateral with angles labeled 1 through 7 and some given angle measures: $\angle 1 = 73^\circ$ and $\angle 5 = 46^\circ$. We nee

1. **Problem Statement:** We are given a kite-shaped quadrilateral EFGD with diagonals intersecting at H. The angle at vertex D, specifically $\angle GDE$, measures 59°. We need to

1. **Problem 1: Complementary Angles** A right angle measures 90°. Two angles are complementary if their sum is 90°.

1. **State the problem:** We need to find the missing angle $\angle U$ in triangle $UVT$ where $\angle V = 50^\circ$ and the exterior angle at $T$ (formed by extending $TP$) is $11

1. **Problem Statement:** We are given two intersecting lines forming four angles. One angle is 39°, the angle opposite it is unknown (?), and two other angles in a triangle are 20

1. **State the problem:** We have two triangles side by side. The left triangle has angles 60° and 65°, and the right triangle has angles 35° and 50°, with the third angle unknown.

1. **State the problem:** We have two intersecting triangles forming an hourglass shape with angles 27°, 35°, 23°, and an unknown angle marked with a question mark opposite the 27°

1. **Problem statement:** Given $XY=32$, $XZ=28$, $JQ=12$, and the radius of the circumscribed circle of $\triangle XYZ$ is $20$, find the length $QK$. 2. **Understanding the probl

Problem: Graph the image of ∆FGH after a dilation with a scale factor of 2 centered at the origin. 1. Formula: For a dilation centered at the origin with scale factor $k$, each poi

1. Statement of the problem: We are given triangle $\triangle STU$ with vertices $S=(-8,-4)$, $T=(-8,8)$, and $U=(4,-8)$ and asked to find the image after a dilation centered at th

1. State the problem: Dilate rectangle RSTU with vertices R(-10,-10), S(10,-10), T(10,5), U(-10,5) by a scale factor of 1/5 centered at the origin. 2. Formula: For a dilation cente

1. **State the problem:** Find the distance between the points $(-3, 6)$ and $(-8, -6)$.\n\n2. **Formula:** The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is giv

1. **State the problem:** Find the distance between the points $(3, 0)$ and $(-3, -8)$.\n\n2. **Formula:** The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is give

1. **State the problem:** Find the distance between the points $(-8, 3)$ and $(-2, -5)$. 2. **Formula used:** The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is g