📐 geometry

1. **State the problem:** We have rectangle EFGH with diagonals intersecting at point I. Given that $GI = 84$ and $HI = -x - 2$, we need to solve for $x$. 2. **Recall properties of

1. **State the problem:** We have a right triangle with legs 20 and 48, and hypotenuse 52. We want to find the angle $x$ opposite the leg of length 20. 2. **Formula used:** To find

1. **Stating the problem:** We have a right triangle with angles 30°, 60°, and 90°, and the side opposite the 30° angle is 10 km. We need to find the length of side $g$, which is o

1. **State the problem:** We need to find the area of a circle given its diameter is 18 inches. 2. **Formula:** The area $A$ of a circle is given by the formula $$A = \pi r^2$$ whe

1. **State the problem:** We need to find the circumference of a circle with radius $r = 5.4$ inches. 2. **Formula:** The circumference $C$ of a circle is given by the formula:

1. **Problem Statement:** Rotate the line segment with endpoints $(-4, -2)$ and $(-2, -1)$ by 90° clockwise about the origin. 2. **Formula for 90° Clockwise Rotation:**

1. **Problem Statement:** Rotate the triangle with vertices at $ (3, -5) $, $ (4, 0) $, and $ (2, 1) $ by 90° counterclockwise about the origin. 2. **Formula for 90° Counterclockwi

1. **Problem statement:** Find the value of $x$ in a right triangle where the legs are 6 and 10, and the hypotenuse is $x$. Then determine if the side lengths form a Pythagorean tr

1. **State the problem:** We need to find the distance across the lake, which is the length of segment $QR$. 2. **Given:**

1. The problem is to find the rule for reflecting the point $(3,13)$ to $(-3,-13)$. 2. Reflection rules depend on the axis or line of reflection. Common reflections include:

1. The problem states that the width $W$ of a rectangle is 5 meters and the length $L$ is 6 meters. The area is given as 30 square meters. 2. The formula for the area $A$ of a rect

1. The problem involves calculating the area and perimeter of squares and rectangles given their side lengths or dimensions. 2. For a square, the area $A$ is given by the formula $

1. The problem asks to identify the term that describes the line segment TR in the triangle with vertices Q, R, S, and T. 2. Given that TR is drawn from vertex T to vertex R, and t

1. **Problem Statement:** Calculate the area and perimeter of a square with side length $s = 52$ inches. 2. **Formulas:**

1. **Problem Statement:** Calculate the area and perimeter of a square with side length $s = 56$ ft. 2. **Formulas:**

1. **State the problem:** We have a regular hexagon with side length 1 cm, and we need to find the value of $x$, which is the length of a vertical segment inside the hexagon.

1. **Problem Statement:** Find the total area of the shaded polygon composed of three parts on the coordinate grid. 2. **Identify the parts:**

1. **Problem statement:** The area of triangle PQR is 20 cm², with sides |PQ| = 10 cm and |PR| = 8 cm. We need to find the two possible values of the angle |∠QPR|. 2. **Formula use

1. **Problem statement:** We have a right triangle with a hypotenuse of length $13\sqrt{2}$, angles $45^\circ$ and $30^\circ$, and a perpendicular dropped from the top vertex to th

1. **State the problem:** We have a rectangular wall divided into three sections: red, green, and blue. The green section is a square with side length 14 feet.

1. **Problem statement:** Find the missing side length of each rectangular solid given the volume. 2. **Formula:** Volume of a rectangular solid is given by $$V = l \times w \times