📐 geometry

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).

1. **State the problem:** Find the perimeter and area of the polygon with vertices S(3,0), T(3,9), U(8,9), and V(8,0). 2. **Identify the shape:** The vertices form a rectangle beca

1. **State the problem:** Find the perimeter and area of the polygon with vertices J(1,2), K(7,2), L(7,8), and M(1,8). 2. **Identify the shape:** The polygon is a rectangle because

1. Problem 49: Find the ratio of the total surface area of three small cubes with side lengths 3 cm, 4 cm, and 5 cm to the surface area of a large cube made by combining their volu

1. **State the problem:** We have rectangle ABCD with perimeter 26.

1. **State the problem:** We have rectangle ABCD with perimeter 26. A half circle with diameter AD is inside the rectangle, and its area is given as $8\pi$. We need to find the per

1. The problem asks for the value of $x + y + z$ where $x$, $y$, and $z$ are central angles in a circle. 2. Recall that the sum of all central angles around a point in a circle is

1. **State the problem:** A man’s eyes are 1.75 m above sea level and he can barely see the top of a lighthouse 20 km away. We need to find the elevation of the lighthouse top abov