📐 geometry

1. **Problem statement:** We have a solid shape made up of a cone and a hemisphere. The radius of both the hemisphere and the cone is $x$ cm, and the perpendicular height of the co

1. **Problem statement:** We have a solid shape made up of a cone and a hemisphere. The radius of both the hemisphere and the cone is $x$ cm, and the perpendicular height of the co

1. **Problem Statement:** We are given an isosceles right triangle with a right angle at the top vertex and two equal sides indicated. We need to find the size of the angle marked

1. **Problem Statement:** A hemispherical bowl of radius 7 cm is full of water. We need to find the area of the wet surface (the inner surface of the hemisphere in contact with wat

1. **Problem Statement:** Given a circle with diameter AB and center O, points A, F, C, B, and D lie on the circumference. CD is perpendicular to AB, and FED is a straight line. An

1. **State the problem:** We are given the vertices of triangle PQR as $P(3,4)$, $Q(7,-2)$, and $R(-2,-1)$. We need to find the equation of the median through vertex $R$. 2. **Reca

1. **Problem statement:** A metallic spherical shell has an external radius of 6 cm and thickness of 1 cm. It is melted and drawn into a wire of diameter 2 cm. If 1 cm of the wire

1. **Problem Statement:** Given that AB and DE are perpendicular to BC, prove that triangles ABC and DEC are similar.

1. It seems you mentioned a circle but did not provide any specific details or equations about it. 2. To help you understand a circle problem, please provide the equation of the ci

1. **Problem Statement:** We want to understand the theorem that states "The angles formed at the same segment of a circle are equal in magnitude." This means if two angles are sub

1. **Problem statement:** Find the point on the second quadrant of a unit circle whose x-coordinate is $-\frac{1}{2}$. The unit circle is defined by the equation $$x^2 + y^2 = 1.$$

1. **State the problem:** Determine if the point $\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$ lies on the unit circle. 2. **Recall the unit circle equation:** The unit ci

1. **State the problem:** We have a right triangle with legs 12.5 m and 22.5 m, and hypotenuse 25 m. We want to find the distance $x$, which is the vertical height from the top cor

1. **State the problem:** We need to determine the space occupied by a water bottle, modeled mathematically. 2. **Identify the shape:** The bottle is roughly a vertical cylinder wi

1. **State the problem:** We have two vertical arrows intersecting a horizontal line, forming angles. The angle labeled "a" is 96° and the angle labeled "b" is (6x - 30)°. We need

1. **Problem Statement:** Find the length of the line segment $MN$ where $M = (5, 3)$ and $N = (1, -2)$. 2. **Formula:** The distance between two points $M(x_1, y_1)$ and $N(x_2, y

1. **Problem Statement:** Determine the number of lines of symmetry in a 7x7 grid pattern with black squares at specified positions. 2. **Understanding Symmetry:** A line of symmet

1. **State the problem:** We need to determine which translation(s) map Figure O onto Figure P on the coordinate plane. 2. **Understand translation:** A translation moves every poi

1. **Problem statement:** We are given a cuboid with base rectangle PQRS and top face W T U V. PR is the diagonal of the base, and M is the midpoint of PR. PU is a vertical edge, a

1. The problem asks to find the complement and supplement of given angles. 2. Recall the definitions:

1. **Problem 1: Find the arc length of a circle with radius 20 cm and central angle $\frac{3}{7}\pi$.** 2. The formula for arc length $s$ is: