📐 geometry

1. **State the problem:** We have a sector $ABCO$ of a circle with radius $R$. The length of the arc $AB$ is given as $\frac{1}{2}R$. We need to find the angle $AOB$ in degrees. 2.

1. The problem appears to involve angles and possibly a variable $u^2$. 2. Since the angles given are $90^\circ$ and $45^\circ$, these might relate to right triangles or angle sums

1. **Problem:** Find the area of the equilateral triangle with side length $5$ and altitude $h = 4\sqrt{15}$. 2. **Formula:** The area $A$ of a triangle is given by $$A = \frac{1}{

1. **State the problem:** We need to find how long it takes to fill a cylindrical jar with water flowing at $20$ ml per second. 2. **Given:** Radius $r = 4$ cm, Height $h = 9$ cm,

1. **State the problem:** We need to find the length (height) of a cylinder given its radius and volume. 2. **Formula for the volume of a cylinder:**

1. **State the problem:** We are given that $SQ$ bisects $\angle PSR$ and that $\angle SPQ \cong \angle SRQ$. We need to prove that $PS \cong SR$. 2. **Understand the given informa

1. **State the problem:** We are given an isosceles triangle $\triangle ABC$ with base $BC$, and perpendiculars $BE$ and $CF$ dropped from vertices $B$ and $C$ onto sides $AC$ and

1. **Problem Statement:** Find the area of the lune formed by two intersecting circles with radii $r$ and $R$, where $R > r$.

1. **Problem Statement:** We have six rays meeting at a common center point, forming adjacent angles around the intersection. Two angles are given: $47.7^\circ$ and $47.9^\circ$, a

1. **Problem Statement:** We have two similar right triangles, a smaller one embedded inside a larger one. Given some side lengths and angles, we need to find the values of the pro

1. **Problem Statement:** We have a rectangle with a shaded region. The right side length is $\frac{1}{4}$ km and the bottom side length is $\frac{2}{3}$ km. We want to find the ar

1. **State the problem:** We are given that $AB \cong BC$, and that $DE$ bisects both $AB$ and $BC$. We need to prove that $BD \cong BE$.

1. **Problem Statement:** Identify which pairs of angles are supplementary angles. 2. **Definition:** Supplementary angles are two angles whose measures add up to $$180^\circ$$.

1. **Problem Statement:** Identify which pairs of angles are vertical angles from the given options. 2. **Definition:** Vertical angles are the pairs of opposite angles made by two

1. **Problem Statement:** Construct a Dragon Curve fractal by following the given iterative geometric steps. 2. **Understanding the Process:** The Dragon Curve is created by repeat

1. **Problem Statement:** Given triangle $\triangle MEC$ with centroid $P$, and lengths $MP=32$, $LP=18.6$, and $CT=45.6$, find the length of segment $LE$. 2. **Key Concept:** The

1. **Problem Statement:** Construct the Dragon Curve fractal by following the given iterative geometric steps. 2. **Understanding the Process:** The Dragon Curve is created by repe

1. **State the problem:** We have a rectangle and a triangle with the same area. The rectangle has height $7$ cm and width $5$ cm. The triangle has height $5$ cm and base $d$ cm. W

1. **State the problem:** We need to find the total surface area of a triangular prism with given side lengths. 2. **Identify the dimensions:** The prism has two triangular bases a

1. **Problem Statement:** Determine if parallelogram $VWXY$ is a rhombus given that the diagonals $WY$ and $XV$ intersect at point $Z$ and the angle between them at $Z$ is $85^\cir

1. **State the problem:** We are given a parallelogram $TUVW$ with diagonals intersecting at point $S$. The segments $ST$ and $SV$ are parts of the diagonals. 2. **Important proper