📐 geometry

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:

1. مسئله: یافتن میانگین کمترین و بیشترین مساحت شکل داده شده است که شامل مستطیل‌های تو در تو با ابعاد مشخص است. 2. تعریف متغیرها و اطلاعات داده شده:

1. The problem asks why the area of a sphere is given by $4\pi r^2$ and not $\frac{4}{3}\pi r^2$, and how this relates to adding infinite circles. 2. First, let's clarify the formu

1. **Stating the problem:** We have a 3D rectangular prism BCDEFGHI with a base BCDE that is a square. The sides BC and DE are given as 12 cm and 8 cm respectively, and the length

1. **State the problem:** Calculate the volume of a prism with a trapezoidal base. 2. **Formula for volume of a prism:**

1. **Problem Statement:** We are given a right triangle VWX with a right angle at X. Points Y and Z are on sides VX and VW respectively, with segment YZ perpendicular to VX. Given

1. **Problem statement:** We have a right triangle EHL with a right angle at E. A segment EP is perpendicular to the hypotenuse HL, with EP = 9.7 units, PH = 18.6 units, and EH = 2

1. **Problem statement:** We have 17 pieces of paper shaped as circular sectors with radius 12 cm, used to form a right circular cone with height $h$. We want to find the maximum v

1. **State the problem:** We need to find the values of angles $x$ and $y$ in the given geometric figure with quadrilateral ABCD, straight line CDE, and isosceles triangle AFB. 2.

1. **Énoncé du problème :** Soient les points $A(-7;1)$, $B(1;5)$, $C(-1;1)$ et les points $E$, $F$, $G$ définis par