📐 geometry

1. **State the problem:** We have a right triangle with one leg measuring 1 cm and the hypotenuse measuring 2 cm. We need to find the length of the other leg. 2. **Formula used:**

1. **State the problem:** Find the value of $x$ and determine if the angles are adjacent or vertical. 2. **Exercise 1:** Angles $120^\circ$ and $x^\circ$ are adjacent on a straight

1. **State the problem:** Find the lengths of the legs and the hypotenuse of the right triangle formed by points $(-5, 3)$ and $(-8, 6)$. 2. **Formula used:** The distance between

1. **Problem Statement:** Calculate the areas of Figure A and Figure B based on the given dimensions and verify the statements about their areas.

1. **Problem 4:** Given two similar figures with corresponding side lengths, find the length $x$. 2. The figures are similar, so corresponding sides are proportional. The formula f

1. **Problem 1: Find the values of $x$ and $y$ in similar triangular prisms.** Given two similar right triangles with sides:

1. **State the problem:** We are given two similar figures (arrows) with known side lengths and need to find the unknown length $x$ in the larger figure. 2. **Recall the property o

1. **Problem:** Parallelogram ABCD is similar to parallelogram EFGH. Given AB = 21 in, EF = 7 in, and AD = 6 in, find the length of GH. 2. **Formula and rules:** For similar parall

1. **State the problem:** We have two similar rectangular prisms with dimensions: - First prism: height = 8 in, length = 12 in, width = 3 in

1. **State the problem:** We have a right triangle with hypotenuse 36 and base 12, and a smaller right triangle inside it with a segment labeled $a$ opposite the hypotenuse side 36

1. **State the problem:** We are given a right triangle with legs of lengths 5 inches and 9 inches, and we need to find the hypotenuse $h$. 2. **Formula used:** For a right triangl

1. **Problem statement:** We have two sectors OWX and OYZ with the same central angle of 75° but different radii: 39 cm and 28 cm respectively. We need to find the area of the shad

1. **State the problem:** We have a sector OAB of a circle with radius $r = 4.3$ mm and area $14$ mm². We need to find the central angle $\theta$ in degrees to 1 decimal place. 2.

1. **State the problem:** We have a rectangle with height 12 in and base 8 in. Inside it, a right triangle is shaded with height $x$ in and base 8 in. The area of the shaded triang

1. **State the problem:** We need to find the height $h$ of a trapezoid given its area and the lengths of the two bases. 2. **Recall the formula for the area of a trapezoid:**

1. **State the problem:** Find the area of trapezoid PKW with parallel sides 1.3 mm and 2.5 mm, and height 0.5 mm. 2. **Formula for the area of a trapezoid:**

1. **State the problem:** We are given a trapezoid with bases of lengths 4 km and 8 km, an area of 42 square kilometers, and we need to find the height $h$. 2. **Formula for the ar

1. **State the problem:** We are given a trapezoid with bases of lengths 4 meters and 12 meters, an area of 48 square meters, and a right side length of 10 meters. We need to find

1. **State the problem:** We are given a trapezoid with bases of lengths 8 inches and 4 inches, an area of 30 square inches, and we need to find the height $h$. 2. **Formula for th

1. **State the problem:** We are given triangle IKL with vertices I(-1, 2), K(0, 3), and L(1, -1). The center of dilation is the origin (0,0) and the dilation rule is given by the

1. **State the problem:** Barbara has a wooden cabin 42 meters wide. She uses 27-meter beams for the roof, which meet at the middle of the cabin's width. We need to find the angle