📐 geometry

1. **Problem statement:** We have a large square ABCD with side length $5$ (since $1 + 3 + 1 = 5$). Inside it, three smaller squares are drawn with side lengths $1$, $3$, and $1$ a

1. **State the problem:** We need to find the radius of a cylinder given its volume and height. 2. **Formula:** The volume $V$ of a cylinder is given by $$V = \pi r^2 h$$ where $r$

1. **Stating the problem:** We have two large triangles sharing a common base of length 10 cm. The left triangle has height 7 cm, and the right triangle has height 12 cm.

1. **Problem statement:** We are given two triangles with angles and sides, and we need to solve for $x$ in the second triangle. 2. **Identify known angles and sides:**

1. **State the problem:** We are given two triangles VWU and YZX with some side lengths and angles, and we need to find the length of side YZ. 2. **Given data:**

1. Problem 6: Determine angle C in a right triangle with sides CB = 5 cm, CA = 13 cm, and angle B = 90°. 2. Use the Pythagorean theorem to find side AB:

1. **State the problem:** We have a right triangle with angles 50°, 90°, and 31°, and one side of length 30 mm opposite the 90° angle. We need to find the missing side $x$ opposite

1. **Problem statement:** We have a right triangle with angles 90° and 45°, and the side opposite the 45° angle is 12 cm. We need to find the missing side $x$, which is adjacent to

1. **Stating the problem:** We need to find the measure of angle $m\angle G$ in quadrilateral $EFGH$. 2. **Given information:**

1. **Stating the problem:** We need to find the measure of angle $\angle G$ in quadrilateral $EFGH$ where $\angle F = 150^\circ$, $EF \cong FG$, and $\angle H$ is a right angle ($9

1. **Stating the problem:** We need to find the length of segment $AB$ in a trapezoid where $DC = 11.5$ units, and $MN = 18.7$ units, with $MN$ parallel to both $AB$ and $DC$. 2. *

1. The problem asks to explain axial (line) symmetry in simple words. 2. Axial symmetry is a geometric transformation where a figure is reflected across a line called the axis of s

1. **State the problem:** We need to find the measures of all angles in an isosceles trapezoid where one angle is given as $82^\circ$. 2. **Recall properties of isosceles trapezoid

1. نبدأ ببيان المشكلة: لدينا ثلاث نقاط $a$, $b$, $c$ ونقطة $i$ هي منتصف القطعة $ab$. المطلوب هو التعبير عن الشعاع $\vec{ac}$ بدلالة الشعاع $\vec{mc}$ و$\vec{ma}$ لأي نقطة $m$. 2. ن

1. نبدأ ببيان المسألة: لدينا ثلاث نقاط $a$, $b$, $c$، و $i$ هو منتصف القطعة $ab$. المطلوب هو إثبات أن الشعاع $\vec{ac}$ يساوي مجموع الشعاعين $\vec{mc}$ و $\vec{ma}$. 2. نعرف أن $i$

1. **Problem Statement:** Given a circle with center $O$, a chord $AB$, and a tangent $PT$ touching the circle at $T$. Line $OC$ is perpendicular to chord $AB$ at point $C$. We nee

1. نبدأ بتحديد المشكلة: نريد إيجاد النقطة $M$ بحيث يكون المتجه $\overrightarrow{CM}$ مساوياً للمتجه $\overrightarrow{AB}$. 2. القاعدة المستخدمة: إذا كان $\overrightarrow{CM} = \ove

1. **State the problem:** We have a solid composed of a hemisphere and a cone joined at their bases. The radius $r$ of both the hemisphere and the cone base is 8 cm. The total surf

1. **State the problem:** We have a solid cylinder of height 11 m and radius 4 m. A cone is removed from the top of the cylinder. The cone has the same base radius 4 m and a slant

1. **Problem Statement:** Given a right-angled triangle with one leg (height) 2.7 m and the other leg (base) 70 cm (0.7 m), and a step pattern along the hypotenuse with steps of 6

1. **State the problem:** We have a parallelogram with angles labeled as follows: top-right angle is $3x + 20$ degrees, bottom-left angle is $2y - 5$ degrees, and bottom-right angl