📐 geometry

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

1. **Find the length of the segment indicated in the top-left circle with two secants:** Given: Outside segments 6 and 8, inner segment unknown (call it $x$).

1. **State the problem:** We have a frustum formed by removing a smaller cone from a larger cone. The larger cone has height 45 cm, the smaller cone removed has height 9 cm and bas

1. The term "scale factor" typically refers to the ratio by which a figure is enlarged or reduced in size in geometry. 2. It is the multiplier used to increase or decrease the dime

1. **Problem 1:** If arcs APB and CQD of a circle are congruent, find the ratio of AB : CD. Since arcs APB and CQD are congruent, their corresponding chords AB and CD are equal in