📐 geometry

1. **Problem Statement:** We have a half-open cylinder with length 455 cm, top diameter 110 cm, and bottom diameter 91 cm. The length is divided into 4 equal parts, and we want to

1. **Problem Statement:** We have a half-open cylinder with length 455 cm, bottom diameter 91 cm, and top diameter 110 cm. The cylinder is cut into 4 equal pieces along its length.

1. The problem is to find the surface area or volume of an open half-cylinder with a given length of 455 cm. 2. To solve this, we need more information such as the radius of the cy

1. **State the problem:** We have a triangle with angles 115°, 35°, and an unknown angle $x$. We need to find $x$. 2. **Formula used:** The sum of the interior angles of any triang

1. **State the problem:** We are given two intersecting lines forming angles, one angle is 80° and the adjacent angle is $x$. We need to find the value of $x$. 2. **Recall the rule

1. **Problem statement:** Given parallelogram ABCD with $DA=4$, $DC=5$, and angle $\angle ADC=60^\circ$. (a) Prove that $\vec{AB} + \vec{AC} = 2\vec{AC}$.

1. **Problem statement:** We are given three points $P(2, -4)$, $Q(-4, -4)$, and $R(-4, 6)$ that lie on a circle. We need to find the quadrant in which the center of this circle li

1. **Problem Statement:** We have a quadrilateral with sides 10 cm (left vertical), 8 cm (top horizontal), 8 cm (right vertical), 12 cm (bottom horizontal), and a diagonal inside t

1. **Problem Statement:** Define and describe right-angled triangles and equilateral triangles. 2. **Right-angled Triangle:** A right-angled triangle is a triangle in which one of

1. **Stating the problem:** We are asked to find the formula for the distance between two points $A(x_1, y_1)$ and $B(x_2, y_2)$.

1. **State the problem:** Given quadrilateral KLMN with KL parallel to NM, LN bisects angle KNM, angle KLN = 54°, and angle MKN = 35°. Find angle KMN. 2. **Understand the propertie

1. **Problem Statement:** We have two triangles with sides $a$, $b$, and $c$. We need to fill in the blanks for the sums of two sides and compare them to the third side.

1. **Problem statement:** Given $m \angle JKM = 100^\circ$, $m \angle NKM = 29^\circ$, and $\overline{KN}$ bisects $\angle LKM$, find $m \angle JKL$. 2. **Understanding the problem

1. **Problem statement:** We need to find the total amount of paint required to paint the top hemispherical part of 3000 Bosu exercise balls. Each ball's top is a hemisphere with a

1. **Problem 16:** Find the height of a right pyramid with volume 72 m³ and base dimensions 800 cm by 300 cm. 2. **Formula:** The volume $V$ of a pyramid is given by $$V = \frac{1}

1. **State the problem:** We have a cylinder with height 24 feet and radius 2 yards. We need to find:

1. **State the problem:** We are given a circle with points P, Q, R, S, and T on its circumference. The angle \(\angle SQR = 52^\circ\) and the angle \(\angle PRT = 16^\circ\). We

1. **Problem Statement:** Find the volumes of the given shapes based on the provided dimensions. 2. **Formulas and Rules:**

1. **Problem statement:** Find the volume of the triangular pyramid (tetrahedron) with a right triangle base of legs 9 km and 9 km, and a height of 11 km. 2. **Formula:** The volum

1. **Problem Statement:** Find the length of the line segment NO where N = (4,0) and O = (12,0) in a square space of dimensions $2a \times 2a \times 2a$.

1. **Problem statement:** We have a right triangle with sides 13, 14, and 15 units, and a semicircle inside the triangle with its diameter along the side of length 14. We need to f