📐 geometry

1. **State the problem:** We need to find the size of angle $k$ given a diagram with intersecting lines and angles $86^\circ$, $78^\circ$, and $102^\circ$. 2. **Recall the rule:**

1. **Problem statement:** We have puzzle pieces with central angles 70°, 32°, 136°, 58°, and one missing piece. These pieces should form a full circle (360°) and a semicircle (180°

1. **State the problem:** We need to construct a line segment AB of length 8 cm and its perpendicular bisector using a ruler and compass. Then, mark point C on the perpendicular bi

1. **State the problem:** We need to find the area of a quadrilateral that is divided into two right triangles by a diagonal. 2. **Identify the triangles and their sides:**

1. **Stating the problem:** We are given a right triangle EFB with perpendicular segments EF and FB, and EB perpendicular to HF. The lengths are $EF=\sqrt{20}$ units and $FH=2$ uni

1. **Problem Statement:** Given points $P(-4,-1)$, $Q(6,3)$, $R(6,b)$, and $S(-4,-3)$, we need to solve several parts related to gradients, parallelism, lengths, midpoints, and pro

1. **Problem Statement:** Prove that a cyclic parallelogram is a rectangle. 2. **Key Definitions:**

1. **Problem Statement:** We have a rectangle with vertical side length $2400$ mm and horizontal side length $2200$ mm. We want to find the length of the diagonal, represented by t

1. **State the problem:** We have a line segment \(\overline{AB}\) with endpoints \(A(1, 3)\) and \(B(5, 3)\).

1. **State the problem:** We have a triangle ABC with vertices A(-4, 3), B(2, 3), and C(-5, 1). We apply a dilation centered at the origin with a scale factor of 5. 2. **Formula fo

1. **Stating the problem:** We have a rectangle with height $2400$ mm and width $2200$ mm. We want to find the length of the diagonals and the point where they intersect. 2. **Form

1. **Problem Statement:** We have a triangle ABC with points A(1,1), B(2,3), C(4,1) that is dilated from the origin to triangle A'B'C' with points A'(3,3), B'(3,9), C'(12,3).

1. **Problem Statement:** We have triangle $\triangle LMN$ with vertices $L(2, 2)$, $M(6, 2)$, and $N(0, 6)$. We want to find the coordinates of $L'$, $M'$, and $N'$ after a dilati

1. **State the problem:** We have a rectangle ABCD with vertices A(2,1), B(2,4), C(6,4), and D(6,1). We need to find the coordinates of A', B', C', and D' after a dilation centered

1. **State the problem:** We have trapezoid ABCD with vertices A(-7,5), B(-7,9), C(-3,9), D(-5,1) and its dilation image A'B'C'D' with vertices A'(-2,1), B'(-2,3), C'(0,3), D'(0,-2

1. **State the problem:** We have trapezoid ABCD with vertices A(-6,6), B(-8,8), C(-2,8), D(-6,2) and its dilated image A'B'C'D' with vertices A'(-2,2), B'(-3,3), C'(-1,3), D'(-2,1

1. **State the problem:** We have triangle MNO with vertices M(-6, 9), N(12, 3), and O(3, -12). It is dilated from the origin to triangle M'N'O' with vertices M'(-2, 3), N'(4, 1),

1. **State the problem:** We have parallelogram WXYZ with vertices W(0,0), X(1,2), Y(4,2), Z(3,0). It is dilated from the origin to form W'X'Y'Z' with vertices W'(0,0), X'(2,4), Y'

1. **State the problem:** We have a line segment \(\overline{PQ}\) with endpoints \(P(0,4)\) and \(Q(-6,4)\).

1. **Problem statement:** We are given three points A, B, and C with C due west of B, the bearing of A from B is 241°, and AC = CB. We need to find the bearing of A from C. 2. **Un

1. **State the problem:** We have rectangle DEFG with vertices D(2,4), E(4,7), F(10,3), and G(8,0). We want to find the coordinates of D', E', F', and G' after a dilation with scal