📐 geometry

1. **Problem statement:** We have a kite-shaped quadrilateral WXYZ with diagonals WY and XZ intersecting at U. 2. **Given:** WY is divided into two segments 12 and 12 by U, so WY =

1. **State the problem:** We have two similar triangles. In the first triangle, angles $\angle L = 55^\circ$ and $\angle J = 100^\circ$ are given. We need to find the measures of a

1. The problem asks to find the perimeter of the given image. 2. The perimeter of a polygon is the sum of the lengths of all its sides.

1. **State the problem:** We need to find the length of segment $YX$ in kite $WXYZ$. 2. **Recall kite properties:** In a kite, two pairs of adjacent sides are equal. Here, $WX = XY

1. **State the problem:** We need to find the value of the unknown angle $x^\circ$ in a figure where angles 43°, 42°, and 120° are given along a horizontal line. 2. **Understand th

1. **State the problem:** We need to find the perimeter of kite WXYZ. 2. **Recall the formula:** The perimeter $P$ of a kite is the sum of the lengths of all its sides. Since a kit

1. **State the problem:** We have kite quadrilateral $WXYZ$ with diagonals intersecting at $U$. Given lengths are $WU=12$, $UY=12$, $XU=6$, and $UZ=24$. We need to find the length

1. **Problem statement:** We have kite $WXYZ$ with angles $m\angle W = 74^\circ$ and $m\angle Y = 56^\circ$. We need to find $m\angle Z$. 2. **Recall kite properties:** In a kite,

1. **State the problem:** We need to find the measure of angle $\angle T$ in trapezoid $TVYZ$ where $TY \cong VZ$ and $TV \parallel YZ$. Given $\angle V = 60^\circ$. 2. **Recall pr

1. **State the problem:** We have triangle LMN with vertices L(1, -2), M(3, 0), and N(2, -3). We want to find the coordinates of the dilated triangle L'M'N' after a dilation with s

1. **Problem:** \nGiven that \(\angle 1\) and \(\angle 2\) are vertical angles, and \(m\angle 1 = (5x + 12)^\circ\), \(m\angle 2 = (6x - 11)^\circ\), find \(m\angle 1\).\n\n2. **Ru

1. **State the problem:** We need to find the range of possible lengths for the third side $n$ of a triangle when the other two sides measure 6 ft and 19 ft. 2. **Recall the triang

1. **State the problem:** Determine if a triangle can be formed with side lengths 0.7, 1.4, and 2.1. 2. **Recall the Triangle Inequality Theorem:** For any triangle with sides $a$,

1. **State the problem:** We are given two vertical angles, \(\angle 1\) and \(\angle 2\), with measures \(m\angle 1 = 5x + 12\) and \(m\angle 2 = 6x - 11\). We need to find the me

1. **State the problem:** We need to find the cosine of angle $S$ in a right triangle $TUS$ where the right angle is at vertex $U$. 2. **Identify the sides relative to angle $S$:**

1. **State the problem:** We need to find the tangent of angle $B$ in a right triangle with vertices $A$, $B$, and $C$. The side lengths given are $AB=17$ units and $BC=8$ units, w

1. **State the problem:** We need to find the horizontal distance of a wheelchair ramp with a vertical rise of 8 inches and an incline angle of 4.76°. 2. **Relevant formula:** In a

1. **State the problem:** We have a stage that is 20 feet long (horizontal length) and rises 2 feet vertically. We want to find the angle of elevation (rake) and check if it is 5°

1. **State the problem:** Find the distance $d(P_1, P_2)$ between the points $P_1 = (2, 5)$ and $P_2 = (-2, -4)$. 2. **Formula used:** The distance between two points $(x_1, y_1)$

1. **State the problem:** Chelsea runs north 0.75 mile, then west 1 mile, and finally runs straight home. We need to find the total distance she runs each week (5 weekdays). 2. **I

1. **State the problem:** John drives south for 8 miles, then east for 15 miles, and returns home by a straight road. We need to find the total distance he drives after 7 days. 2.