📐 geometry

1. **Problem Statement:** We are given a circle with center at point $T$. Lines $TU$, $TV$, $TW$, and $TX$ divide the circle into sectors. Two sector angles at $T$ are given as $75

1. **Problem statement:** We have a shape formed by joining triangle ABC and sector CBD of a circle with radius 7 cm and center C.

1. The problem states that there are 2 angles labeled as $a$ and the exterior angle is 35 degrees. 2. Recall the exterior angle theorem: The exterior angle of a triangle is equal t

1. **State the problem:** We have an isosceles triangle with two base angles each labeled $a^\circ$ and the vertex angle labeled $x^\circ$. The two sides are extended upwards formi

1. **State the problem:** Calculate the volume $V$ of a cone using the formula given. 2. **Formula:** The volume of a cone is given by

1. مسئله را بیان می‌کنیم: در شکل داده شده، با توجه به اندازه‌های دوای متجانس، باید تفاوت شعاع‌های کوچکتر را پیدا کنیم و نسبت محیط مثلث کوچکتر به محیط مثلث بزرگ‌تر را محاسبه کنیم.

1. **Problem statement:** Find the total area of the shaded regions formed by two concentric circles with radii 3 cm and 6 cm, and a minor sector AOB with angle 60°.

1. **Problem statement:** Calculate the percentage of the area of the circle that is shaded, given the areas of triangles and sectors in the circle with diameters AC and BD interse

1. **Stating the problem:** We need to find how many liters of paint are required to paint 10,000 wooden blocks. Each block consists of a triangular prism and a square part with gi

1. **Problem statement:** a. Given triangle ABC with vertices A(3,0), B(6,4), and C(-1,3), show that ABC is a right-angled isosceles triangle.

1. **Problem Statement:** Given triangle ABC with vertices A(-4,6), B(-1,2), and C(0,5), we need to find the images of ABC under various transformations: central symmetry about (1,

1. **Stating the problem:** We are given a parallelogram ABCD with points E and F on sides AD and DC respectively. We know that $m(DAF) = m(FAB)$ and $m(CBE) = m(EBA)$, and the len

1. Problem: ABCD is a parallelogram with DE \perp AB, |KE|=3 cm, |AE|=4 cm, |KC|=25 cm. Find |EB|. 2. Important properties: In a parallelogram, opposite sides are equal and paralle

1. The problem is to understand the key difference between the ASA (Angle-Side-Angle) and AAS (Angle-Angle-Side) triangle congruence criteria. 2. Both ASA and AAS are used to prove

1. The problem is to find the total surface area of a trapezoidal prism with given dimensions. 2. The formula for the surface area of a prism is the sum of the areas of all its fac

1. **Problem statement:** We have a pentagon composed of a rectangle and an isosceles triangle on top. The rectangle has height $27$ and the triangle has two equal sides of length

1. The problem is to find the total surface area of a rectangular prism with dimensions 5.5 cm, 19.2 cm, and 4 cm. 2. The formula for the surface area $A$ of a rectangular prism wi

1. **Problem statement:** The circle C touches the y-axis at point A(0,3) and passes through point B(2,7). 2. **Find the equation of the perpendicular bisector of AB.**

1. **Stating the problem:** We need to find the value of the expression $$\frac{a(\triangle ABE)}{a(\triangle CDE)} \times \frac{m(\angle ABE)}{m(\angle DCE)}$$ where $a(\triangle)

1. **Stating the problem:** We are given a right triangle ABC with a right angle at A. Point D lies on segment CB such that AD is perpendicular to CB. The area of triangle ADC is 1

1. **State the problem:** We need to find the size of angle $y$ adjacent to a $20^\circ$ angle on a straight line. 2. **Recall the rule:** Angles on a straight line add up to $180^