📐 geometry

1. **Problem Statement:** We have a right triangle $\triangle ACE$ with sides $AC=12$, $CE=16$, and $EA=20$. Points $B$, $D$, and $F$ lie on sides $AC$, $CE$, and $EA$ respectively

1. **Problem statement:** We have an equilateral triangle $\triangle ABC$ with side length 4. Point $M$ is the midpoint of $AC$, and $C$ is the midpoint of $BD$. We need to find th

1. **Problem Statement:** We are given a circle with center $O$ and a tangent line $AC$ at point $C$ on the circle. The segment $AB$ has length 8, and the radius $OC$ has length 5.

1. **Problem 5:** One angle of a quadrilateral is 135°. Work out the size of the other angles. A quadrilateral has four angles that add up to 360°.

1. **Problem Statement:** Given triangle $\triangle ABC$ with $AB < AC$, incentre $I$, and incircle $\omega$ touching sides $BC, AC, AB$ at points $D, E, F$ respectively. Point $P$

1. **Problem statement:** We need to find the value of the angle $x$ formed at the center $O$ of a regular octagon by two lines extending from $O$ to two adjacent vertices. 2. **Ke

1. সমস্যাটি হলো: একটি ফাঁপা লোহার গোলকের বাহিরের ব্যাসার্ধ 14 সে.মি. এবং লোহার বেধ 2 সে.মি. দেওয়া আছে। এর ভিতরের নিরটে গোলক তৈরি করা হয়েছে যা একটি ঘনক আকৃতির বাক্সে ঠিকভাবে ফিট করে

1. ප්‍රශ්නය: අපට දී ඇති ත්‍රිකෝණ තුනක් (a) 6 cm, 8 cm, 10 cm, (b) 4.5 cm, 6 cm, 7.5 cm, (c) 5 cm, 5 cm, 4 cm යන පාර්ශව දිග මත පදනම්ව ත්‍රිකෝණ නිර්මාණය කර, ඒවායේ කෝණ එකතුව 180° බව ස

1. Problem: Explain what the "trojkat egipski" (Egyptian triangle) is. 2. The Egyptian triangle is a right triangle with side lengths in the ratio 3:4:5.

1. **Problem statement:** We have a solid shape made up of a cone and a hemisphere. The radius of both the hemisphere and the cone is $x$ cm, and the perpendicular height of the co

1. **Problem statement:** We have a solid shape made up of a cone and a hemisphere. The radius of both the hemisphere and the cone is $x$ cm, and the perpendicular height of the co

1. **Problem Statement:** We are given an isosceles right triangle with a right angle at the top vertex and two equal sides indicated. We need to find the size of the angle marked

1. **Problem Statement:** A hemispherical bowl of radius 7 cm is full of water. We need to find the area of the wet surface (the inner surface of the hemisphere in contact with wat

1. **Problem Statement:** Given a circle with diameter AB and center O, points A, F, C, B, and D lie on the circumference. CD is perpendicular to AB, and FED is a straight line. An

1. **State the problem:** We are given the vertices of triangle PQR as $P(3,4)$, $Q(7,-2)$, and $R(-2,-1)$. We need to find the equation of the median through vertex $R$. 2. **Reca

1. **Problem statement:** A metallic spherical shell has an external radius of 6 cm and thickness of 1 cm. It is melted and drawn into a wire of diameter 2 cm. If 1 cm of the wire

1. **Problem Statement:** Given that AB and DE are perpendicular to BC, prove that triangles ABC and DEC are similar.

1. It seems you mentioned a circle but did not provide any specific details or equations about it. 2. To help you understand a circle problem, please provide the equation of the ci

1. **Problem Statement:** We want to understand the theorem that states "The angles formed at the same segment of a circle are equal in magnitude." This means if two angles are sub

1. **Problem statement:** Find the point on the second quadrant of a unit circle whose x-coordinate is $-\frac{1}{2}$. The unit circle is defined by the equation $$x^2 + y^2 = 1.$$

1. **State the problem:** Determine if the point $\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$ lies on the unit circle. 2. **Recall the unit circle equation:** The unit ci