📐 geometry

1. **State the problem:** We have a segment \(\overline{QR}\) that is dilated by a scale factor of 2 to produce \(\overline{Q'R'}\). The length of \(\overline{Q'R'}\) is given as 7

1. **Problem statement:** A figure containing angle $\angle FGH$ is dilated by a scale factor of 5 to form a new figure containing angle $\angle F'G'H'$. Given that $\angle FGH = 1

1. **State the problem:** We have a line segment IJ with length 2 units. It is dilated by a scale factor of $\frac{3}{4}$ to form a new segment I'J'. We need to find the length of

1. **State the problem:** We have a line segment VW with length 26 units. It is dilated by a scale factor of $\frac{1}{4}$ to form a new segment V'W'. We need to find the length of

1. Let's find the surface area and volume of the first shape: a right cylinder with two hemispheres on its ends. 2. The cylinder has radius $r=6.0$ cm and height $h=8.5$ cm. Each h

1. **Problem Statement:** We have triangle $\triangle ABC$ divided into six smaller triangles by lines drawn from each vertex through a common interior point. Four of these smaller

1. **Problem xi:** Two bricks with dimensions $8\times44\times10$ cm and $16\times22\times10$ cm are placed face-to-face to form three different rectangular prisms. Two prisms are

1. **Problem statement:** We have three points: P, Q, and R. Q is 60 km due north of P, and R is 45 km due east of Q. We need to find the bearing of R from P.

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