📐 geometry

1. **Problem statement:** Given parallel lines TY || HT and transversals BG and KP, with angles \(m\angle KXY = (15x - 2)^\circ\), \(m\angle TWP = (7x + 6)^\circ\), \(m\angle MRH =

1. **Problem statement:** The perimeter of triangle $\triangle PQR$ is 94 units, and $QS$ bisects $\angle PQR$. We need to find the lengths of $PS$ and $RS$. 2. **Relevant theorem:

1. The problem asks to find the length of side $x$ in parallelogram $EFGH$ given the sides of parallelogram $ABCD$ and some sides of $EFGH$. 2. We know that parallelograms that are

1. **State the problem:** We have a circle with center $D$, radius $CD = 17$, and a central angle $\angle CDE = 162^\circ$. We need to find the area of sector $CDE$. 2. **Formula f

1. **State the problem:** We have a circle with center F, points E and G on the circumference, and the angle \(\angle EFG = 54^\circ\). Given \(EF = 13\) units, find the length of

1. **State the problem:** We have a circle with center $Q$, points $P$ and $R$ on the circumference, and the angle $\angle PQR = 134^\circ$. Given $PQ = 7$ units, find the length o

1. **Stating the problem:** We are given a sector of a circle with a central angle of 320°. 2. **Formula used:** The fraction of the circle represented by the sector is given by th

1. The problem states that the figure represents $\frac{1}{5}$ of a full circle, and we need to find the measure of the marked central angle. 2. Recall that a full circle measures

1. **State the problem:** We need to find the total area of a polygon composed of a trapezoid on the left and a right triangle on the right, sharing a horizontal base. 2. **Identif

1. **State the problem:** We need to find the total cost of wax to cover the floor, which is an irregular polygon composed of a rectangle and a right triangle. 2. **Identify the sh

1. **State the problem:** We need to find the area of a rhombus with one diagonal measuring 16 cm and each side measuring 15.5 cm. 2. **Recall the formula for the area of a rhombus

1. **State the problem:** We need to find the area of a trapezoid with parallel sides of lengths 4 m and 7 m, and a height of 6.2 m. 2. **Formula for the area of a trapezoid:**

1. **Problem statement:** Given a circle centered at $O$, with tangent segments $VY$ and $VW$ from point $V$ to the circle, and segment $YW$ passing through $O$. Given $VY=8.7$ and

1. **State the problem:** We need to find the total area of the given polygon with sides labeled 16 ft, 22 ft, 4 ft, and 8 ft. 2. **Analyze the figure:** The polygon can be divided

1. **State the problem:** We are given triangle \(\triangle PQR\) with perimeter 94 units, and \(QS\) bisects \(\angle PQR\). We need to find the lengths \(PS\) and \(RS\). 2. **Re

1. **State the problem:** We are given a rectangle WXYZ with area 115.5 in\(^2\). We need to find the perimeter of triangle \(\triangle XYZ\).\n\n2. **Identify known information:**

1. **Problem statement:** We have a 30-60-90 right triangle with the side opposite the 30° angle equal to 12, and we need to find the length of side $b$, which is adjacent to the 3

1. **State the problem:** Given a right triangle with legs $a$ and $b$ and hypotenuse $c$, where $a = 2\sqrt{3}$ and $c = 2b$, find the length of side $b$.

1. **State the problem:** We have a right triangle with legs $a$ and $b$, and hypotenuse $c$. Given $c=13$ and $a=12$, find the length of side $b$. 2. **Formula used:** According t

1. The problem asks which theorem(s) explain why a rectangle is a parallelogram. 2. A rectangle is a quadrilateral with four right angles, meaning each angle measures 90 degrees.

1. **State the problem:** Ahmed recalls a theorem that states if one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelo