📐 geometry

1. **State the problem:** We are given the equation of a circle $$(x-1)^2 + (y+4)^2 = 25$$ and need to find its centre and radius. 2. **Recall the standard form of a circle's equat

1. **State the problem:** We are given that $\overrightarrow{MP}$ bisects $\angle LMN$. We need to prove that $$2 \times m\angle LMP = m\angle LMN$$.

1. **State the problem:** We are given the statement: "If two angles are vertical, then they are ruined."

1. **State the problem:** We are given the statement: "If two angles are supplementary, then the sum of their measures is one hundred eighty."

1. **Problem statement:** Points P and Q are on opposite sides of a lake, R is on land, M is the midpoint of RP, and N is the midpoint of RQ. The segment MN is measured as $(6k + 8

1. **State the problem:** We have a solid made of a cylinder on top of a hemisphere. The total volume is given as $$\frac{2684}{3} \pi$$ cm$^3$. The radius $r$ of both the cylinder

1. **Problem Statement:** We are given a shape consisting of a large blue rectangular prism with a smaller white rectangular prism on top. A dashed segment labeled $l$ is on the to

1. **State the problem:** We need to find the area of a composite polygon consisting of a top parallelogram-like section and a bottom trapezoid-like section. 2. **Identify the shap

1. **State the problem:** We have three parallel lines $l \parallel m \parallel n$ cut by a transversal. We are given an angle of $137^\circ$ at the middle intersection (above line

1. **State the problem:** We need to find the equation of the line containing all possible images of point $C(5,1)$ after a dilation centered at $O(2,0)$ with any scale factor. 2.

1. **State the problem:** We need to find the perimeter of $\triangle CED$, which is the sum of the lengths of its three sides: $CE$, $ED$, and $DC$. 2. **Identify given lengths:**

1. **Problem Statement:** We are given a circle with center $A$, a tangent line $\overleftrightarrow{BC}$ touching the circle at point $B$, with $AB = 7$ and $BC = 24$. We need to

1. **State the problem:** We need to find the coordinates of point $G$, the bottom-left vertex of rectangle $DEFG$, given that $C$ is the center of the rectangle, $D$ is the top-le

1. **State the problem:** We have a rectangle DEFG with center C at coordinates $C(8,7)$ and one vertex $E(20,25)$. We need to find the coordinates of vertex $G$. 2. **Recall the p

1. **State the problem:** We need to reflect a polygon with vertices $(1,-1)$, $(1,-5)$, $(8,-8)$, $(9,-3)$, and $(5,-3)$ over the line $y = x - 1$. 2. **Reflection formula:** To r

1. **State the problem:** We need to reflect a quadrilateral with vertices at $(-9, 4)$, $(-8, 2)$, $(-5, 1)$, and $(-1, 8)$ over the line $$y = x + 1$$.

1. **State the problem:** We need to reflect a figure over the line $$y = x + 3$$ and plot the reflected points. 2. **Reflection formula:** To reflect a point $(x, y)$ over the lin

1. The problem asks to identify the final answer or the reflected points of a light blue pentagon. 2. Reflection in geometry means flipping a shape over a line (called the line of

1. **State the problem:** We need to reflect two pentagons over the line $$y = x + 3$$ and plot the reflected points. 2. **Reflection formula:** To reflect a point $(x,y)$ over the

1. **State the problem:** We need to reflect the polygon with vertices $(-8, 6)$, $(-7, 9)$, $(-3, 6)$, $(-4, 2)$, and $(-7, 4)$ over the line $y = x + 3$. 2. **Formula and approac

1. **State the problem:** We need to find the area of a circle with radius $27$ mm. 2. **Formula:** The area $A$ of a circle is given by the formula: