📐 geometry

1. **Problem Statement:** We have two internally tangent circles. The smaller circle has radius $12$ cm and lies inside the larger circle with radius $18$ cm. The smaller circle's

1. **State the problem:** We have a right triangle with hypotenuse length 29 feet. One leg is 19 feet less than twice the other leg.

1. **State the problem:** We are given two supplementary angles \(\angle C\) and \(\angle D\) with measures \(m\angle C = 12x - 7\) and \(m\angle D = 16x + 19\). We need to find th

1. The problem states that \(\angle 1\) and \(\angle 2\) are vertical angles, and we are given \(m\angle 1 = 7x + 1\) and \(m\angle 2 = 5x + 47\). We need to find \(m\angle 2\). 2.

1. **Problem Statement:** Identify the type of angle pairs for the given angle pairs in the diagram. 2. **Important Definitions:**

1. **Problem Statement:** Identify the type of angle pairs for the given angle pairs in the diagram with two parallel lines and a transversal. 2. **Recall definitions:**

1. **State the problem:** We need to find the volume of a concrete cylindrical pillar with diameter 4 meters and height 5 meters, then calculate the total cost of the concrete give

1. **Problem statement:** Calculate the surface area of a cube placed on top of a right triangular prism. The prism has a rectangular base of 15 cm by 6 cm and a right triangle on

1. **State the problem:** We want to find what fraction of the volume of a 1-inch cube is occupied by one smaller cube with edge length 1/3 inch. 2. **Formula for volume of a cube:

1. **State the problem:** We want to find how many $\frac{1}{3}$-inch cubes fit inside a rectangular prism with dimensions 3 inches (height), 2 inches (length), and $1 \frac{2}{3}$

1. **State the problem:** Find the length of segment $XY$ given that $X$ is the midpoint of segment $WY$, with $WX = 3x - 1$ and $WY = 10x - 26$.

1. **Problem statement:** We have a triangle with sides 14 and 10, and an angle of 15° opposite the unknown side $x$. We need to find $x$. 2. **Formula used:** We use the Law of Co

1. **State the problem:** We need to find the length $x$ of the hypotenuse in a right triangle where one leg is 23 m and the angle adjacent to the base is 59°. 2. **Identify the si

1. **Problem Statement:** Determine if parallelogram ABCD is a rectangle given: - $AB = 4x + y$

1. The problem involves checking calculations related to trigonometric functions (sin, cos, tan) and the Pythagorean theorem. 2. The Pythagorean theorem states that in a right tria

1. **Problem Statement:** Draw and label a vector $\overrightarrow{CD}$ intersecting a line segment $\overline{AB}$, with plane P containing $\overline{AB}$ but not $\overrightarro

1. **Problem statement:** Find the height of triangle ABC where sides AB and BC are 10 units each, base AC is 8 units, and a perpendicular from B to AC divides AC into two equal pa

1. **Stating the problem:** Given that the measure of angle $X$ is 75 degrees and the measure of an exterior angle at $Z$ is 160 degrees, we want to find what is true about the val

1. The problem asks to describe the changes in shape, size, position, and orientation of triangle ABC when it is reflected to triangle A'B'C'. 2. Reflection is a transformation tha

1. Problem 8: Find the area of each given figure. Since the figures are described as a right triangle and a right trapezoid, we use the following formulas:

1. **State the problem:** Find the value of $x$ in each of the given figures. 2. **Figure (1):** The triangle has sides 5, 3, and $x$, with a base length 11.