📐 geometry

1. The problem asks to identify the triangle congruence criteria shown in the diagram with two right triangles sharing a hypotenuse and a leg. 2. The key congruence criteria for ri

1. **State the problem:** Prove that triangles $\triangle LMN$ and $\triangle PON$ are congruent. 2. **Given:**

1. **State the problem:** Given that $E$ is the midpoint of segments $GN$ and $WQ$, prove that $WN \cong GQ$. 2. **Understand the given information:** Since $E$ is the midpoint of

1. **Problem statement:** Given quadrilateral ABCD with AB = 9, BC = 10, AB ∥ DC, angle B = 63°, and a perpendicular height from C to AD with foot length 4, find the perimeter of A

1. **State the problem:** We have a quadrilateral ABCD with AB parallel to DC, AB = 9, BC = 10, AD = 4, and angle B = 63°.

1. **State the problem:** Find the area of parallelogram ABCD given the base and height. 2. **Formula:** The area $A$ of a parallelogram is given by:

1. **State the problem:** Find the area of rectangle GHIJ given the side lengths. 2. **Recall the formula for the area of a rectangle:**

1. **State the problem:** Find the scale factor given the corresponding coordinates of points before and after scaling. 2. **Formula:** The scale factor $k$ is found by dividing th

1. The problem asks whether the dilation of Figure A to Figure B is an enlargement or a reduction. 2. Dilation is a transformation that changes the size of a figure by a scale fact

1. **Problem:** Translate rectangle QRST with vertices Q(-6, -1), R(-3, 1), S(1, -5), and T(-2, -7) by the rule $(x, y) \to (x + 5, y + 7)$. 2. **Formula:** To translate a point $(

1. **Problem statement:** We need to find the length of the red line segment in a complex polygon composed of triangles with given side lengths 12, 4, 1, 15, and 1 (red segment). 2

1. **State the problem:** We need to find the tangent of angle $W$ in a right triangle with sides opposite and adjacent to $W$ given as 12 and 37 respectively. 2. **Recall the form

1. **Problem statement:** Find the cosine of angle $X$ in a right triangle $XVW$ where the right angle is at vertex $V$. The sides are $XV=36$, $XW=85$, and $VW=77$. 2. **Recall th

1. **State the problem:** Find the cosine of angle $F$ in right triangle $FDE$ where $\angle D$ is the right angle. 2. **Recall the cosine definition:** For an angle in a right tri

1. **State the problem:** We need to find the cosine of angle $T$ in right triangle $TUV$ where $TU=20$, $UV=48$, and hypotenuse $TV=52$. 2. **Recall the cosine definition:** In a

1. **State the problem:** We have a quadrilateral with two known angles: 96° and 110°, and two missing angles labeled 1 and 2. We need to find the values of angles 1 and 2. 2. **Re

1. **State the problem:** We are given the coordinates of triangle UVW and need to find the slopes and lengths of its sides, then classify the triangle.

1. **Stating the problem:** We have a circle with four inscribed angles measuring 80°, 32°, 25°, and three unknown angles labeled $a$, $b$, and $c$ opposite the known angles. We wa

1. 문제를 이해하기: 점 $P(a,0)$에서 원 $x^2 + y^2 - 4x - 2y + 4 = 0$에 그은 접선의 길이가 2일 때, 양수 $a$의 값을 구하는 문제입니다. 2. 원의 중심과 반지름 구하기: 주어진 원의 방정식을 완전제곱식으로 변형합니다.

1. **State the problem:** We need to find expressions for the perimeter ($P$) and area ($A$) of three polygons with right angles, given side lengths involving $x$.

1. **Stating the problem:** We are given an L-shaped figure with a horizontal segment labeled $x - 1$ and a vertical segment labeled $x$. The height of the vertical segment is 2, a