📐 geometry

1. **Stating the problem:** We are given a circle with center O and points A, B, C on the circle. The triangle OAB is formed such that \(\angle OAB = 20^\circ\) and \(\angle OCB =

1. **Problem:** Find the length of arc EF in a circle with radius 12 m and central angle 120°. 2. **Formula:** The length of an arc $L$ is given by

1. **Problem statement:** We want to express the side length $s$ of a square as a function of its diagonal length $d$. Then, express the area $A$ of the square as a function of $d$

1. **Problem statement:** We have a trapezium ABCD with parallel sides AB and CD.

1. **بيان المسألة:** لدينا رباعي ABCD حيث (AB) يوازي (CD) و4D = BC، وM منتصف الضلع [BC]. الزوايا المعطاة هي $\angle AMB = 24^\circ$ و $\angle CMD = 66^\circ$. المطلوب هو قياس الزاو

1. **Problem:** Find the distance between points $A(0,0)$ and $B(3,4)$. 2. **Formula:** The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the distance for

1. The problem asks for examples of a triangular prism. 2. A triangular prism is a three-dimensional solid with two parallel triangular bases and three rectangular faces connecting

1. **Stating the problem:** Calculate the surface area of a triangular prism given the formula $$SA = (s_1 + s_2 + s_3)h + bh$$ where $s_1, s_2, s_3$ are the sides of the triangula

1. **Stating the problem:** We have a triangle with a base of length 20 and a horizontal segment inside it parallel to the base with length 8. The vertical sides are divided into s

1. **State the problem:** We need to find the total surface area of a hemisphere with diameter 11 meters. 2. **Recall the formula:** The surface area of a sphere is given by $$4\pi

1. **State the problem:** We have a rectangular tank with length $3$ m and width $2$ m. It is filled with $2.7$ m$^3$ of water without overflowing. We need to find the height $h$ o

1. **Problem statement:** We have a star made of 6 identical quadrilaterals. Each quadrilateral has an angle $x$ at the star's points and two adjacent angles of $35^\circ$. We need

1. Problem: Find the missing side length $x$ in the right triangle with legs $16\sqrt{3}$ and $16$, and hypotenuse $x$. 2. Formula: Use the Pythagorean theorem for right triangles:

1. **Problem statement:** Calculate the bearing of point Y from point Z given the angles at points X and Y relative to north. 2. **Understanding bearings:** A bearing is measured c

1. The problem asks to verify the correctness of the given angle measures in two figures involving parallel lines and transversals. 2. For the first figure with angles a=83°, b=97°

1. The problem states that in parallelogram $ABCD$, angle $D$ is a right angle, and we want to prove that the diagonals are congruent. 2. Bao suggests proving this by using the fac

1. **State the problem:** Find the degree measure of the angle marked with an arc at the bottom of the figure, given that the angles are supplementary and some angles are labeled a

1. The problem asks to identify the location of the origin on the coordinate plane. 2. The origin in a coordinate plane is the point where the x-axis and y-axis intersect.

1. **State the problem:** We need to find the perimeter of a composite figure made of a rectangle and a semicircle attached to the right side of the rectangle. 2. **Identify given

1. **State the problem:** Determine if triangles TUS and GIH are similar. 2. **Recall similarity criteria:** Triangles are similar if their corresponding angles are equal or their

1. **State the problem:** We have two rectangles, one inside the other. The outer rectangle has dimensions 12 units by 9 units.