📐 geometry

1. **State the problem:** We have two intersecting triangles forming an hourglass shape with angles 27°, 35°, 23°, and an unknown angle marked with a question mark opposite the 27°

1. **Problem statement:** Given $XY=32$, $XZ=28$, $JQ=12$, and the radius of the circumscribed circle of $\triangle XYZ$ is $20$, find the length $QK$. 2. **Understanding the probl

Problem: Graph the image of ∆FGH after a dilation with a scale factor of 2 centered at the origin. 1. Formula: For a dilation centered at the origin with scale factor $k$, each poi

1. Statement of the problem: We are given triangle $\triangle STU$ with vertices $S=(-8,-4)$, $T=(-8,8)$, and $U=(4,-8)$ and asked to find the image after a dilation centered at th

1. State the problem: Dilate rectangle RSTU with vertices R(-10,-10), S(10,-10), T(10,5), U(-10,5) by a scale factor of 1/5 centered at the origin. 2. Formula: For a dilation cente

1. **State the problem:** Find the distance between the points $(-3, 6)$ and $(-8, -6)$.\n\n2. **Formula:** The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is giv

1. **State the problem:** Find the distance between the points $(3, 0)$ and $(-3, -8)$.\n\n2. **Formula:** The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is give

1. **State the problem:** Find the distance between the points $(-8, 3)$ and $(-2, -5)$. 2. **Formula used:** The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is g

1. **State the problem:** Find the distance between the points $(8, -4)$ and $(3, 8)$. 2. **Formula:** The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by

1. **State the problem:** Find the distance between the two points $(-4,7)$ and $(7,-5)$ on the coordinate plane. 2. **Formula used:** The distance $d$ between two points $(x_1,y_1

1. The problem is to find the distance between the two points $(3, -3)$ and $(7, -7)$ on the coordinate plane. 2. We use the distance formula between two points $(x_1, y_1)$ and $(

1. **State the problem:** We are given a triangle with a mid-segment parallel to the base. The top side is labeled $4x$, the mid-segment is labeled 14, one side is labeled $7x - 9$

1. **State the problem:** Find the distance between the two points $(-6, 8)$ and $(9, -4)$ on the Cartesian plane. 2. **Formula used:** The distance $d$ between two points $(x_1, y

1. **State the problem:** Find the distance between the points $A(5,5)$ and $B(8,1)$ on the Cartesian plane. 2. **Formula used:** The distance $d$ between two points $(x_1,y_1)$ an

1. The problem asks to find the distance between two points $(1,1)$ and $(-7,-8)$ on the Cartesian plane. 2. We use the distance formula between two points $(x_1,y_1)$ and $(x_2,y_

1. **Problem Statement:** A plane cuts through a rectangular pyramid passing through three of its lateral faces and is perpendicular to the base. We need to determine the shape of

1. **Problem Statement:** We have a rectangular pyramid and a plane that cuts through the pyramid. The plane passes through the apex (vertex) of the pyramid and is perpendicular to

1. **State the problem:** Find the distance between the two points $(4, 5)$ and $(-3, -2)$ on the coordinate plane. 2. **Formula used:** The distance $d$ between two points $(x_1,

1. **State the problem:** We have a composite figure made of a parallelogram stacked on top of a square. The parallelogram has a base of 5 cm and a height of 12 cm. The square has

1. **Problem Statement:** Given a right triangle PQR with right angle at P (\(\angle QPR = 90^\circ\)), \(PQ = 6\) cm, and point M on hypotenuse PR such that \(\angle QPM = \angle

1. **State the problem:** We have triangle $ABC$ with $\angle CAB = 82^\circ$ and $\angle C = 68^\circ$. The perpendicular bisector of side $AB$ intersects $AB$ at $D$ and $BC$ at