📐 geometry

1. **Problem statement:** Find the area swept by a door that opens as a sector of a circle with radius $80$ cm and central angle $45^\circ$.

1. The problem is to show the diagram of a circle. 2. A circle is defined as the set of all points in a plane that are at a fixed distance (radius) from a fixed point (center).

1. **Problem statement:** Prove that the diameter of a circle which bisects a chord also bisects the angle subtended by the chord at the center of the circle. 2. **Given:** A circl

1. The problem is to find the eccentricity $e$ of a conic section. 2. The eccentricity $e$ is defined as the ratio of the distance from any point on the conic to the focus, divided

1. **Problem statement:** We have two similar statues A and B. The volume of statue B is 20% less than the volume of statue A. We need to find the percentage decrease $k$ in the su

1. **State the problem:** We have a kite with sides labeled as $n$ cm, $n$ cm, $n+3$ cm, and $n+3$ cm. The perimeter is given as 110 cm. We need to find the value of $n$. 2. **Form

1. **State the problem:** We have a right triangle with sides labeled as $x + 2$ cm, $\frac{x}{2} + 4$ cm, and $x + 5$ cm. The perimeter is given as 36 cm. We need to find the valu

1. **Problem statement:** We have a cube ABCDEFGH with side length 6 cm. Points T and V are midpoints of edges AB and CH respectively.

1. **Problem statement:** We have a cube ABCDEFGH with side length 6 cm. T is the midpoint of edge AB, and V is the midpoint of edge CH.

1. **State the problem:** Calculate the size of the largest angle in the triangle with angles $x + 72^\circ$, $4x - 18^\circ$, and $x$. 2. **Use the triangle angle sum rule:** The

1. **Stating the problem:** We need to find the values of angles $x$ and $y$ given the angles $44^\circ$, $52^\circ$, and $97^\circ$ in a geometric figure involving points $A$, $B$

1. The problem is to find the values of angles $x$ and $y$ given the angles $44^\circ$, $52^\circ$, $y^\circ$, $x^\circ$, and $97^\circ$ in a geometric figure. 2. We use the rule t

1. **State the problem:** In triangle $\triangle ART$, segment $SM$ is parallel to $AT$. Given some side lengths, find the indicated missing lengths using properties of similar tri

1. **Problem statement:** In triangle $\triangle ART$, line segment $SM$ is parallel to $AT$. Given some side lengths, find the indicated missing lengths using properties of simila

1. **Determine the distance between points I (14, 8) and J (8, -3).** The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is:

1. **Problem:** Find the measure of each numbered angle in the triangle TWR with a right angle at W and angles 60° and 30°. 2. **Formula and rules:** The sum of the interior angles

1. **Problem statement:** Given lines AC and DE are parallel, and DC is perpendicular to both AC and DE. Segment BD has length $\frac{4}{3}$. We need to find the length of side DE.

1. **Stating the problem:** Given points A(2,2), B(4,0), and C(-2,0), find the area of triangle ABC. 2. **Formula used:** The area of a triangle with vertices at coordinates $(x_1,

1. **State the problem:** We need to find a series of transformations that map polygon ABCDE with vertices near (1,3), (1,2), (2,1), (3,1), (3,3) onto polygon A'B'C'D'E' with verti

1. **State the problem:** We need to find the area of a square given its vertices at approximately (1.5, 3), (2, 6), (5, 5), and (4, 2.5). 2. **Recall the formula for the area of a

1. **Problem statement:** Given kite ABCD with BN = 15 and AB = 17, find the length of diagonal BD. 2. **Understanding the kite properties:** In a kite, two pairs of adjacent sides