📐 geometry

1. **State the problem:** We have a right circular cylinder with length (height) $h = 7.4$ cm and diameter $d = 2.6$ cm. We want to find the volume of the cylinder in cubic centime

1. **State the problem:** We have a right triangle AOB with right angle at O, OA = 12 cm, OB = 5 cm. We need to find angle BAO and the area of the shaded region between two arcs ce

1. **State the problem:** Melur walks along the path A to B, then B to C, then C to A. We know distances AB = 3.6 km, AC = 8.4 km, and bearings: B from A is 054°, C from B is 132°.

1. **State the problem:** Find the distance between the points $(-12,1)$ and $(12,-1)$.\n\n2. **Recall the distance formula:** The distance $d$ between two points $(x_1,y_1)$ and $

1. The problem is to find the distance between the points $Y(-3,5)$ and $X(7,1)$.\n\n2. Use the distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$: $$d = \sqrt{(x_2 -

1. **State the problem:** We need to find the distance between the two points $(-3, 5)$ and $(7, -1)$ on the coordinate plane. 2. **Recall the distance formula:** The distance $d$

1. The problem is to find the distance between the points $(-3,5)$ and $(7,1)$ using the distance formula. 2. The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is

1. **State the problem:** Find the distance between the points $(-2,5)$ and $(4,-3)$.\n\n2. **Recall the distance formula:** The distance $d$ between two points $(x_1,y_1)$ and $(x

1. **Problem 1:** Find the radius of a circle with area 24 cm² and sector BAC area 3 cm². 2. The area of a circle is given by $$A = \pi r^2$$.

1. **Problem 26:** Find the radius $r$ of a circle with area 24 cm$^2$ and sector BAC area 3 cm$^2$. 2. The area of a circle is given by $$A=\pi r^2.$$ Given $$24=\pi r^2,$$ solve

1. **State the problem:** We are given a triangle ABC with sides AC = 7 cm, BC = 10 cm, and angle BAC = 65°. We need to find the size of angle ABC to the nearest 0.1°. 2. **Identif

1. Let's start by stating the problem: We need to create and analyze three types of circle problems involving secants and tangents: intersecting circles, secant-secant, and secant-

1. **Problem statement:** Given a circle with center O, points A, B, and C lie on the circle. The angle at A is $2x + 15^\circ$ and the angle $\angle OBC = x$. We need to express t

1. Problem: Find the area of the lower base of a frustum of a regular pyramid given volume $V=93.3333$ m³, upper base dimensions $2.5 \times 4$ m, and altitude $h=4$ m. Step 1: Cal

1. **State the problem:** We have three regular nonagons (9-sided polygons) A, B, and C. Polygons A and B share a side. We need to find the sum of the angles $x$ and $y$ formed bet

1. **Problem statement:** Given a circle with center O and points A, B, C, D, E, F on the circumference or inside, with angles $\angle AEB = 50^\circ$, $\angle EBC = 80^\circ$, and

1. **Problem 1: Find the length $x$ of the sector given the area is 46 cm$^2$.** The area $A$ of a sector of a circle is given by the formula:

1. **Problem:** Prove that \(\triangle ABC \cong \triangle FED\) given \(\angle A = 70^\circ\), \(\angle C = 50^\circ\), \(BC = 10\) cm, and \(\angle E = 60^\circ\), \(\angle F = 7

1. **Problem Statement:** Prove that triangles $\triangle ABC$ and $\triangle FED$ are congruent, and determine if triangles $\triangle SRT$ and $\triangle DEF$ are similar, provid

1. Problem a) Calculate the area of the shaded sector with central angle $\alpha = 30^\circ$ in a circle of radius 4. 2. The area of a sector is given by the formula:

1. **State the problem:** Find the area of a trapezoid with bases and height given. 2. **Identify the bases and height:** The trapezoid has two parallel sides (bases) of lengths 7.