📐 geometry

1. **State the problem:** We are given two intersecting lines forming vertical angles. One angle measures 152° and we need to find the measures of angles $x$, $y$, and $z$. 2. **Re

1. **State the problem:** We are given the volume of a sphere as $36000\pi$ units³ and need to find its radius. 2. **Formula for the volume of a sphere:**

1. **Problem:** Find the radius of a cone-shaped funnel that holds 314 cubic inches of water with a height of 12 inches. Use 3.14 for pi. 2. **Formula:** The volume of a cone is gi

1. **Problem statement:** We have two rectangular prisms (storage containers) with given dimensions and need to find missing measurements.

1. **State the problem:** We have two rectangular prisms (crates) with given dimensions and need to find the volume of the first and the height of the second.

1. **Problem statement:** Given triangle ABC with BC parallel to DE, and segments BD = 2x - 4, DC = 3x + 6, BE = x + 5, and EA = 8x - 5, find the value of $x$. 2. **Formula and rul

1. **State the problem:** We need to show that the median from the right angle vertex $C$ to the hypotenuse $AB$ in a right triangle is half the length of the hypotenuse. 2. **Reca

1. **State the problem:** Find the area of the composite shape consisting of two rectangles and a right triangle arranged horizontally. 2. **Identify the shapes and their dimension

1. **Square with diagonal $3\sqrt{2}$:** The formula relating the diagonal $d$ and side length $s$ of a square is $$d = s\sqrt{2}$$

1. **Problem Statement:** We want to determine if triangle $\triangle ABC$ with vertices $A(5,5)$, $B(-3,-1)$, and $C(1,-3)$ is a right triangle by calculating the lengths of its s

1. **State the problem:** We need to find the length of side $OP$ in rectangle $OPRS$ given angles and side lengths. 2. **Given:** $\angle R = 90^\circ$, $\angle S = 90^\circ$, $\a

1. **State the problem:** We need to find an expression for the area of the shaded region, which is the area of the circle minus the area of the rectangle inside it. 2. **Identify

1. **State the problem:** We need to find the area of triangle $\triangle DEF$ where sides $d = 990$ cm, $e = 990$ cm, and the included angle $\angle F = 141^\circ$. 2. **Formula u

1. **Постановка задачи:** Дана окружность с четырьмя точками на ней: A, B, C, D. Отрезки между точками имеют длины в отношении $AB : BC : CD : AD = 3 : 4 : 5 : 6$. Нужно найти угол

1. **Stating the problem:** We are given two diagrams with marked angles at points P and Q. We need to identify angles that are alternate, cointerior, and corresponding to the mark

1. **State the problem:** Draw a triangle with a 90° angle and one side measuring 5 centimeters. 2. **Important concepts:** A triangle with a 90° angle is a right triangle.

1. **State the problem:** We need to find the bearing of point D from point C using the protractor scale. 2. **Understanding bearings:** Bearings are measured clockwise from the no

1. **State the problem:** We need to find the bearing of point B from point A using the given semicircle protractor. 2. **Understanding bearings:** Bearings are measured clockwise

1. **Problem statement:** We need to find the length of the fence BD and the area of the quadrilateral ABCD with given sides and angles. 2. **Given data:**

1. **Problem Statement:** We have a quadrilateral ABCD with sides AB = 246 m, BC = 312 m, AD = 257 m, and angles \(\angle DÂB = 96^\circ\) and \(\angle BĈD = 78^\circ\). We need to

1. **Problem 1(a):** Calculate $x$ in a right triangle with legs 4 cm and 6 cm, hypotenuse $x$. 2. Use the Pythagorean theorem formula: $$x^2 = a^2 + b^2$$ where $a$ and $b$ are le