📐 geometry

1. **State the problem:** We have an isosceles triangle with two equal acute angles and one obtuse angle. The obtuse angle is 2.5 times one of the acute angles. We need to find the

1. The problem asks to identify which line segment is drawn in the figure based on the description. 2. The figure is a circle with center at point $X$.

1. The problem states that angles 1 and 2 are supplementary. 2. Supplementary angles are two angles whose measures add up to 180 degrees.

1. **Problem:** Compare the lengths of segments MR and RP in triangle MRP where angle R is 21° and angle P is 19°. 2. **Rule:** In any triangle, the side opposite the larger angle

1. **Problem:** Compare the segments MR and RP in triangle MRP with angles 21° at R and 19° at P. 2. **Formula and rule:** In any triangle, the side opposite the larger angle is lo

1. **State the problem:** We are given a triangle ABC with points A(-3,0), B(-2,3), and C(-1,1) and its translated image A'B'C' with points A'(0,1), B'(0,4), and C'(2,0). We need t

1. **Problem statement:** Given points A, B, C, D, E, F lying alternately on lines $\ell$ and $\mathcal{M}$, with $AB \parallel ED$ and $FE \parallel BC$, show that $$\frac{|OA|}{|

1. **Problem Statement:** Show that if line segment PR is not parallel to BC in triangle ABC, then the ratio $\frac{|AP|}{|AB|} \neq \frac{|AR|}{|AC|}$. Then conclude that for any

1. **Problem Statement:** Given a triangle ABC with points P on AB and Q on AC such that the line segment PQ is parallel to BC, and the ratio $\frac{|AP|}{|PB|} = \frac{|AQ|}{|QC|}

1. **State the problem:** We have a composite figure made of a parallelogram, a trapezoid, and a triangle with given dimensions. We need to find: - The combined area of the paralle

1. **State the problem:** We need to find the equation of a circle representing the perimeter of a circular park with center at $(2, -3)$ and a point on the edge at $(4, -1)$. 2. *

1. **State the problem:** We need to find the area of a composite figure made up of a parallelogram, a trapezoid, and a triangle with given dimensions.

1. **State the problem:** We need to find the combined area of the parallelogram and the triangle in the composite figure. 2. **Recall formulas:**

1. **State the problem:** We need to find the image of a triangle after reflecting it across the y-axis and then translating it by the vector $v=(-1,0)$. The original triangle has

1. **State the problem:** Given a circle with points T, S, R, Q on its circumference and angles $m\angle STQ = 246^\circ$ and $m\angle TSR = 123^\circ$, find: (a) $m\angle STQ$ (al

1. **Problem statement:** Given a circle with a triangle inscribed such that one angle is $46.8^\circ$ and the angle $x$ is on the opposite side of the diameter line, find the valu

1. **Stating the problem:** We are given two intersecting lines forming angles, with one angle labeled $x$ and the corresponding angle on the opposite side labeled $72^\circ$. We n

1. **State the problem:** We have a right triangle where the longer leg is 7 cm longer than the shorter leg, and the hypotenuse is 9 cm longer than the shorter leg. We need to find

1. **State the problem:** We have a right triangle with hypotenuse 13, one leg 7, and the other leg $x$. We need to find $x$ rounded to the nearest hundredth. 2. **Formula used:**

1. **State the problem:** We are given three vectors \(\overrightarrow{AB}\), \(\overrightarrow{CD}\), and \(\overrightarrow{CE}\) originating from point \(C\) with angles \(82^\ci

1. **State the problem:** Reflect the point $(2,0)$ across the line $y=x$. 2. **Formula and rule:** When reflecting a point $(a,b)$ across the line $y=x$, the coordinates swap plac