📐 geometry

1. **State the problem:** Given triangles \(\triangle ART \sim \triangle FIN\) with altitudes \(AS = 4\) and \(FE = 3\), and sides \(AT = x+1\) and \(FN = x-1\), find the lengths o

1. **State the problem:** We have parallelogram KLMN with sides LM = 5 cm, MN = 3 cm, and diagonal LN = 3.2 cm. We need to find the measure of angle LKN to the nearest degree (Part

1. Problem a: Find $x$ where vertical angles are equal and adjacent angles sum to 180°. 2. Use the rule: adjacent angles on a straight line sum to 180°.

1. **Problem statement:** Calculate the missing angle in each diagram where all angles at a point add up to 360°. 2. **Formula:** Sum of angles at a point = 360°.

1. The problem involves understanding the transformation of a point on a figure in the coordinate plane. 2. We are given an original point $(3, 5)$ and its image after transformati

1. **State the problem:** We need to find which arcs have the same size central angle given their arc lengths and radii. 2. **Formula:** The central angle $\theta$ (in radians) for

1. The problem asks for the radian measure of angle \(\angle A\) in a circle with radius 3, where the angle is marked as \(\pi\) radians. 2. The radian measure of an angle in a cir

1. **State the problem:** Calculate the surface area of a gas tank composed of a cylinder with a hemisphere on each end.

1. **State the problem:** We need to find the surface area of a shape made by joining a hemisphere on top of a cylinder. The hemisphere and cylinder share the same radius $r=9$ cm,

1. **State the problem:** Find the distance between the points (7, 9) and (2, -3) using the right triangle formed by these points and the point (2, 9). 2. **Formula used:** The dis

1. The problem is to find the angle $\angle AOB$, not $\angle ACB$.\n\n2. To find $\angle AOB$, we need to understand the points $A$, $O$, and $B$ and their positions or vectors. U

1. **Problem statement:** We have two towers represented by two circles with radii 100 km and 150 km. A common external tangent line CE touches the circles at points C and E. We wa

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