📐 geometry

1. مسئله: در متوازی الاضلاع ABCD طول BH را بیابید. 2. ابتدا باید بدانیم که BH عمود بر کدام ضلع است. در متوازی الاضلاع، ارتفاع از یک رأس به ضلع مقابل عمود است.

1. **Problem statement:** We need to find the area of the shaded sector of a circle where the central angle $\theta = 56^\circ$ and the circumference $C = 31.416$ cm. 2. **Find the

1. **State the problem:** Lee wants to paint the front face and top face of a cuboid. The front face measures 54 cm by 32 cm, and the top face measures 32 cm by 28 cm. The paint co

1. **State the problem:** We are given a circle with points A, B, C, and D on its circumference. Triangle ABD has an angle at vertex A of 56° and triangle BCD has an angle at verte

1. **State the problem:** We have a triangle with sides 5 km, 12 km, and 13 km. We need to find the greatest angle between the roads. 2. **Identify the greatest angle:** The greate

1. Problem 1: Given a circle with points P, Q, R, S on the circumference and angles 28° at R and 20° adjacent to x at S, find x using the theorem that angles in the same segment ar

1. **State the problem:** We have a sector OAB of a circle with center O and radius 6 cm. The length of the arc AB is given as $5\pi$ cm.

1. **State the problem:** We are given a right circular cone with volume $71,148\pi$ cubic centimeters and base area $5,929\pi$ square centimeters. We need to find the slant height

1. The problem is to draw two triangles connected by vertical projection lines with given measurements: $\Delta x = 60$ mm, $\Delta y = 65$ mm, $\Delta z = 40$ mm, and $\psi = 30^\

1. The problem states that the volume of a cube is 125 cm\(^3\). We need to find its total surface area. 2. Recall the formulas:

1. **State the problem:** We are given a prism with a cross-sectional perimeter of 36 cm and a cross-sectional area of 54 cm². The length of the prism is 15 cm. We need to find the

1. **State the problem:** We are given a prism with a lateral surface area of 175 cm² and a cross-sectional area of 40 cm². We need to find the total surface area of the prism. 2.

1. The problem states that the lateral surface area and the cross-sectional area of a prism are both 175 cm². 2. The lateral surface area (LSA) of a prism is the perimeter of the b

1. **State the problem:** We are given a prism with a cross-sectional perimeter of 48 cm and a length of 10 cm. We need to find the lateral surface area of the prism. 2. **Recall t

1. **State the problem:** We have a scalene triangle ABC with sides BC = 85.94, AC = 74, and an angle of 45 degrees at vertex B. We want to find the length of side AB and the heigh

1. **State the problem:** We need to find the length of the diagonal $AG$ in the cuboid, given angles $27^\circ$ and $42^\circ$ at vertex $A$, and the base edge $AD = 63$ mm. 2. **

1. **State the problem:** We need to find the length of the edge $AG$ in the cuboid, given angles $27^\circ$ and $42^\circ$ at vertex $A$, and the length $AD = 63$ mm. 2. **Analyze

1. **Problem Statement:** Given two isosceles triangles $\triangle ABC$ and $\triangle DBC$ on the same base $BC$, with vertices $A$ and $D$ on the same side of $BC$, and $AD$ exte

1. **State the problem:** Calculate the area of a quadrilateral with sides $AB=708$, $BC=584$, $CD=706$, $DA=609$, and diagonals $AC=915$, $BD=931$. 2. **Approach:** We can split t

1. The problem asks for the total surface area of two garage doors. 2. The left door has a width of $2.5$ m and a height of $2$ m.

1. **Problem Statement:** Construct quadrilateral ABCD with given sides and diagonals: AB = 6.8 cm, AD = 5.8 cm, AC = 8.9 cm, BD = 7.3 cm, BC = 4.1 cm. 2. **Step 1: Draw diagonal A