📐 geometry

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

1. The problem is to construct a rectangle that can be divided into four identical squares. 2. To solve this, we need to understand that four identical squares arranged together fo

1. **Problem statement:** Construct a rectangle that can be divided into four identical squares. 2. **Understanding the problem:** A rectangle divided into four identical squares m

1. Vấn đề: Chứng minh góc $\angle FDO = 90^\circ$. 2. Công thức và quy tắc: Để chứng minh góc vuông, ta cần chứng minh rằng hai đoạn thẳng tạo thành góc đó vuông góc, tức là tích v

1. **Đề bài:** Cho đường tròn $(O)$ và điểm $A$ nằm ngoài đường tròn. Từ $A$ vẽ hai tiếp tuyến $AB$, $AC$ với tiếp điểm $B$, $C$. Vẽ đường kính $BD$ của đường tròn $(O)$; $AD$ cắt

1. **Problem statement:** Find the lengths of $DC$ and $BD$ in the given figure where $BD$ is perpendicular to $BA$ and $DC$ is parallel to $BA$. 2. **Given:**

1. **Problem Statement:** Find the area of a triangle with base $b$ and height $h$. 2. **Formula:** The area $A$ of a triangle is given by the formula:

1. مسئله: رسم محور x و y در دستگاه مختصات دکارتی. 2. محور x (محور افقی) خطی است که از چپ به راست کشیده شده و مقدار y در آن صفر است.

1. **Problem Statement:** A rectangular pond has a length of 75 metres and a breadth of 50 metres. A uniform bank of width 4 metres is built all around the outside of the pond. Ins

1. The problem is to find the sum of two angles: 45 degrees and 60 degrees. 2. The formula for the sum of angles is simply addition: $$\text{Sum} = \theta_1 + \theta_2$$

1. সমস্যাটি হলো একটি বৃত্তের সমীকরণ $b(x,y) = x^2 + y^2 = 4$ এবং একটি বিন্দু $(4,3)$ এর অবস্থান নির্ণয় করা। 2. বৃত্তের সমীকরণ $x^2 + y^2 = r^2$ যেখানে $r$ হলো বৃত্তের ব্যাসার্ধ। এ

1. **Énoncé du problème :** Vérifier la véracité des affirmations vectorielles données et écrire leurs réciproques en précisant si elles sont vraies ou fausses.

1. **State the problem:** We have a triangle with angles labeled as follows: the top vertex angle is 80 degrees, the left base angle is $2x - 6$, and the right base angle is $3x +

1. **Stating the problem:** We need to find the surface area of a rectangular prism (cuboid) with length $24$ cm, width $12$ cm, and height $18$ cm.