📐 geometry

1. **Problem statement:** We have a square ABCD with side length 9. A circle centered at A with radius AD is drawn, and another circle with diameter DC is drawn. These two circles

1. **Problem statement:** Given an equilateral triangle ABC with side length $2\sqrt{a}$, a point P inside the triangle satisfies $PA=2$, $PB=2\sqrt{3}$, and $PC=4$. We need to fin

1. **Problem Statement:** Given an equilateral triangle $ABC$ with side length $2\sqrt{a}$, a point $P$ inside the triangle satisfies $PA=2$, $PB=2\sqrt{3}$, and $PC=4$. We need to

1. **Problem Statement:** We have a triangle ABD with a smaller equilateral triangle DEC inside it. We know angle $\angle A = 80^\circ$ and need to find the size of angle $\angle A

1. **Problem statement:** Given a circle with chord $AB$ of length 16 units, $D$ is the midpoint of chord $AB$, and $C$ is the midpoint of arc $ACB$. The length $CD$ is 4 units. We

1. **Stating the problem:** We have a quadrilateral C D E G formed by a parallelogram C D F G and a triangle D F E. (a) We need to identify the type of quadrilateral C D E G.

1. The problem asks to write the coordinates of the vertices of the given figures. 2. For Triangle ABC, the vertices are given as:

1. The problem states that the area of a rectangle is 24 square meters. 2. The formula for the area of a rectangle is $$\text{Area} = \text{length} \times \text{width}$$.

1. **Problem statement:** We have a square of side length 15 cm with 4 identical quarter circles drawn inside it, one in each corner. The shaded part is the region inside the squar

1. **Đề bài:** Cho hình chóp $S.ABCD$ với đáy $ABCD$ là hình bình hành tâm $O$. $N$ là trung điểm của $SC$, $M$ nằm trên cạnh $AD$ sao cho $AD=3AM$. Tìm giao điểm $K$ của đường thẳ

1. Дано: вписанный четырёхугольник с углами, где нужно найти углы $x$ и $y$. 2. Вспомним важное свойство вписанного четырёхугольника: сумма противоположных углов равна 180°. То ест

1. The problem involves finding the areas of two parallelograms given their heights and base lengths. 2. The formula for the area of a parallelogram is $$\text{Area} = \text{base}

1. **Énoncé du problème :** Soient les points $A(1,-1)$, $B(-1,1)$, $C(\sqrt{3},\sqrt{3})$ et le cercle $(C)$ de centre $A$ passant par $C$. Nous devons :

1. **State the problem:** We need to find which figure has a greater area: a parallelogram with base 16.5 yd and height 15 yd, or a square with side length 15.1 yd. 2. **Formula fo

1. **State the problem:** We need to find which square has a greater area given their side lengths. 2. **Formula for the area of a square:**

1. **State the problem:** Calculate the area of two shapes: a parallelogram with base 16 yd and height 17.4 yd, and a rectangle with base 13.9 yd and height 19 yd. 2. **Formulas:**

1. **State the problem:** We need to find which figure has a greater area: a rectangle with dimensions 17 mm by 9 mm or a square with side length 13.2 mm. 2. **Formula for area:**

1. **State the problem:** We need to determine which figure has a greater area: a rectangle with length 16 mm and width 8 mm, or a parallelogram with base 13 mm and height 12 mm. 2

1. **State the problem:** We need to find the height of a parallelogram given its area and base. 2. **Formula:** The area $A$ of a parallelogram is given by the formula:

1. **Problem statement:** Show that $$\frac{2b}{c+a-b} = \frac{\cot \frac{C}{2} + \cot \frac{A}{2}}{\cot \frac{B}{2}}$$ for triangle ABC with sides $a,b,c$ opposite angles $A,B,C$

1. مسئله: میانگین کمترین و بیشترین مساحت شکل را پیدا کنید، با توجه به اینکه اختلاف مساحت مستطیل‌های ABCD و EFGH دو برابر مساحت ناحیه مشترک آن‌ها است. 2. تعریف متغیرها و روابط: