📐 geometry

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.

1. **State the problem:** We need to find the volume of a solid composed of a right pyramid mounted on a rectangular block. Both have a square base of side length 6 cm. The block h

1. The problem is to find the area of a trapezoid with parallel sides measuring $1 \frac{1}{2}$ cm and $3 \frac{1}{2}$ cm, and a height of 4 cm. 2. Convert the mixed numbers to imp

1. **State the problem:** We have a floor with vertices at J(2,1), K(2,8), L(9,8), and M(9,1) on a coordinate plane. We need to identify the shape and find its perimeter and area.

1. **State the problem:** We have a solid made of a rectangular block with a square base 6 cm by 6 cm and height 10 cm, with a right pyramid on top having the same square base and

1. The problem states that a triangle is drawn with vertices A(5, 1), B(7, 6), and C(1, 3). 2. The graph shows point A plotted at (5, 5) instead of (5, 1).