📐 geometry

1. **Problem statement:** We are given a circle with a tangent line \( \overrightarrow{GH} \) touching the circle at point \( I \), and a chord \( IJ \). The angle between the tang

1. **Problem statement:** Given a circle with two tangent lines \( \overrightarrow{EF} \) and \( \overrightarrow{ED} \) touching the circle at points F and D respectively, and the

1. **Problem Statement:** Find the measure of the minor arc $JN$ given two secants intersecting outside the circle at point $L$ with an angle of $41^\circ$ between them.

1. **Problem statement:** We are given a circle with points R, T, S on its circumference and a tangent line PQ⃗ touching the circle at point R. 2. The angle between the tangent lin

1. **Problem statement:** We need to find the value of angle $x$ in a right triangle with a smaller isosceles triangle nested inside it. The smaller triangle has an angle of $48^\c

1. The problem is to understand and apply the Pythagorean theorem, which relates the lengths of the sides of a right triangle. 2. The Pythagorean theorem states that in a right tri

1. **Problem:** Find the height of a rectangular prism given the lateral area and the perimeter of the base. 2. **Formula used:** The lateral area $L$ of a rectangular prism is giv

1. **State the problem:** We are given three points $F(2,9)$, $G(-7,0)$, and $H(-1,1)$ that form a triangle. We need to find the perimeter of this triangle. 2. **Formula used:** Th

1. The problem asks for the equation Jasmine can use to find the area of a parallelogram cut from a 4 x 6 inch index card. 2. The formula for the area of a parallelogram is:

1. **Calculate the area of the triangle** The problem: Find the area of a triangle with base 12 cm and height 8 cm.

1. **State the problem:** We need to find the area of a sector of a circle with radius 5 cm and central angle 60°. 2. **Formula for the area of a sector:** The area $A$ of a sector

1. **State the problem:** We need to find the area of the yellow shaded region, which is the area of the square minus the area of the inscribed circle. 2. **Identify given values:*

1. **State the problem:** Given triangle ABC with BD as the bisector of segment AC, prove that triangle ABD is congruent to triangle CBD using a flowchart proof. 2. **Identify give

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