📐 geometry

1. Problem 49: Find the ratio of the total surface area of three small cubes with side lengths 3 cm, 4 cm, and 5 cm to the surface area of a large cube made by combining their volu

1. **State the problem:** We have rectangle ABCD with perimeter 26.

1. **State the problem:** We have rectangle ABCD with perimeter 26. A half circle with diameter AD is inside the rectangle, and its area is given as $8\pi$. We need to find the per

1. The problem asks for the value of $x + y + z$ where $x$, $y$, and $z$ are central angles in a circle. 2. Recall that the sum of all central angles around a point in a circle is

1. **State the problem:** A man’s eyes are 1.75 m above sea level and he can barely see the top of a lighthouse 20 km away. We need to find the elevation of the lighthouse top abov

1. གྲུ་བཞི་དང་པང་གསུམ་གྱི་ཁ་ཕྱོགས་ཀྱི་འགྲུབ་འབྲས་ག་རེ་ཡིན་ཞིག་བསྐྱར་བཤད་གནང་བ་དགོས། 2. གྲུ་བཞི་དང་པང་གསུམ་གྱི་ཁ་ཕྱོགས་ནི་གྲུ་བཞི་གི་ཁ་ཕྱོགས་དང་པང་གསུམ་གྱི་ཁ་ཕྱོགས་ཡིན། དེ་ནི་གྲུ་བཞ

1. **State the problem:** We are given a rectangular prism with points A, B, C, D on the front face and E, F, G, H on the back face. We know the lengths CD = 2 cm, DE = 3 cm, and B

1. **State the problem:** We have two similar cylinders. The smaller cylinder has a volume of 800 cm^3 and a height of 10 cm. The larger cylinder has a height of 15 cm. We need to

1. **State the problem:** We are given a rectangular prism with points A, B, C, D, E, F, G, H and lengths CD = 65 cm, DE = 63 cm, and BD = 85 cm. We need to find the length of the

1. **State the problem:** We are given a circle with points D, C, A, and B. Point C lies on the vertical segment from D to A inside the circle. We need to find the angle $x^\circ$

1. **Problem statement:** Given a rectangular cuboid with points A, B, C, D on the front face and E, F, G, H on the back face, and given lengths CD = 20 cm, DE = 60 cm, and BD = 40

1. The problem is to plot the point $P(3, 4, -4)$ in a three-dimensional coordinate system. 2. The point $P$ has coordinates $x=3$, $y=4$, and $z=-4$.

1. The problem asks to identify which labelled angles on the triangular prisms are right angles. 2. A right angle measures exactly $90^\circ$.

1. نبدأ بتحديد طول السبلاج شاكي من الشكل المعطى، حيث طول السبلاج هو مجموع الأضلاع التي تحتوي على جذور. 2. الأضلاع المعطاة هي:

1. The problem asks to identify which labelled angles among a, b, e, f, and g on two triangular prisms are right angles. 2. A right angle measures exactly 90 degrees.

1. نبدأ بقراءة المسألة: مساحة قاعدة المبتدع هي 675 دامج مربع، وطول القاعدة ثلاث أضعاف عرضها. 2. نفرض أن عرض القاعدة هو $x$، إذن طول القاعدة هو $3x$.

1. **State the problem:** We need to prove that triangle ACB is isosceles, meaning two of its sides are equal in length. 2. **Identify the points:** Let the coordinates of points A

1. The problem states that $AH = HB$ and $CH$ is perpendicular to $AB$. We need to analyze the geometric properties given these conditions. 2. Since $AH = HB$, point $H$ is the mid

1. **Problem 1:** Given triangle $ABC$ with point $H$ on $AB$ such that $AH \perp AB$ and $CH \perp AB$. We want to understand the properties of $\triangle ACB$ and the point $H$.

1. The problem involves finding the measure of angle $b$ in a triangle where the external angles at $a$ and $c$ are given as $65^\circ$ and $132^\circ$ respectively. 2. Recall that

1. **State the problem:** We have two intersecting diagonal lines crossing a horizontal line, forming angles labeled as follows: the upper left and right angles next to the horizon