📐 geometry

1. **State the problem:** We have a 30°-60°-90° right triangle with hypotenuse length 10. We need to find the lengths of sides $x$ (opposite 60°) and $y$ (opposite 30°). 2. **Recal

1. **Stating the problem:** We have a right triangle with legs of lengths $x$ and 10, and hypotenuse $2x + 21$. We need to find the value of $x$. 2. **Formula used:** By the Pythag

1. **State the problem:** Find the equation of one median of the triangle with vertices A(2,3), B(8,1), and C(5,7), and find the point of concurrency (centroid) of the medians. 2.

1. **State the problem:** We need to find the lengths of the bases $AB$ and $DC$ of the isosceles trapezoid $ABCD$ to calculate its perimeter.

1. **State the problem:** We have a frustum formed by removing a smaller square pyramid from a larger similar square pyramid. The base edge length of the frustum is 36 mm, and the

1. **Problem statement:** We need to find the size of angle $k$ given that another angle in the figure is $81^\circ$. 2. **Assumption:** Since the problem involves angles and one a

1. **State the problem:** We have a right triangle with angles 30°, 60°, and 90°. The side opposite the 30° angle is given as $14\sqrt{3}$ km, and we need to find the hypotenuse $q

1. The user asked to draw circles, but no specific equation or parameters for the circles were provided. 2. To draw a circle, the general equation is $$ (x - h)^2 + (y - k)^2 = r^2

1. **Problem statement:** We have a right triangle with angles 45°, 45°, and 90°. One leg is 3 cm, and the other leg is $g$. We need to find $g$ in simplest radical form. 2. **Impo

1. **State the problem:** We have a right triangle with angles 30°, 60°, and 90°. The side opposite the 60° angle is 1 meter, and we need to find the side $k$ opposite the 30° angl

1. **State the problem:** We need to find the area of a ring-shaped region (annulus) with an inner radius of 15 ft and a width of 5 ft. 2. **Identify the radii:** The inner radius

1. **State the problem:** We have a circle with center C and radius 7 m. The central angle \(\angle DCE\) measures 100°. We need to find the length of the major arc \(dFE\), which

1. **State the problem:** We are given a triangle UVW with angles $\angle U = 30^\circ$, $\angle V = 104^\circ$, and $\angle W = 46^\circ$. Side $UV = 11$ units is opposite $\angle

1. The problem involves identifying angle relationships formed by two parallel lines cut by a transversal. 2. We use the definitions of angle pairs:

1. **Problem Statement:** Given a right triangle with a base of 16, a vertical side labeled $n$, a hypotenuse labeled $m$, and an angle of $30^\circ$ opposite side $n$, find the va

1. Problem 14: Given a right triangle with a 30° angle and sides labeled $m$, $n$, and 16, find the correct values of $m$ and $n$. 2. Recall the properties of a 30°-60°-90° triangl

1. **Problem statement:** Given a right triangle with hypotenuse $6\sqrt{3}$ and an angle of $30^\circ$, find the lengths of sides $x$ (adjacent to $30^\circ$) and $y$ (opposite to

1. **Stating the problem:** We have a right triangle with a 45° angle, the side opposite this angle is 6, the hypotenuse is $m$, and the adjacent side is $n$. We need to find $m$ a

1. **Problem statement:** Given a right triangle with a hypotenuse of length $5\sqrt{2}$ and one angle of $45^\circ$, find the lengths of the legs $u$ and $v$. 2. **Formula and rul

1. The problem states that in a parallelogram, two adjacent interior angles are given as $(7a - 4)^\circ$ and $(4a + 47)^\circ$. 2. Important property: Adjacent angles in a paralle

1. **State the problem:** We need to find the value of $x$ such that quadrilateral ABCD is a parallelogram. 2. **Given:** Angles at vertices A and C along diagonal AC are $(12x + 1