📐 geometry

1. **Problem:** Find the equation of the circle with center $(-1, 2)$ and diameter $6$. 2. **Formula:** The equation of a circle with center $(h, k)$ and radius $r$ is:

1. **Problem statement:** We are given a parallelogram with diagonals of lengths 28 and 30, and one side length of 13. We need to find the other side length and the area of the par

1. **Problem statement:** We have a trapezoid with diagonals of lengths 6 and 10, and the segment connecting the midpoints of the bases (the mid-segment) is 4. We want to find the

1. **Stating the problem:** We need to find a sequence of rigid transformations and dilations that takes square EFGH to square ABCD.

1. The problem asks to identify sequences of transformations that show triangles ABC and AED are similar, given AC = 6 units. 2. Similar triangles can be shown by transformations s

1. The problem asks to identify the invalid step in the proof that all isosceles triangles are similar. 2. The proof attempts to show similarity by applying a sequence of transform

1. Problem: Determine if two line segments or two angles can be similar. 2. Two line segments are similar if they have the same shape and their lengths are proportional. Since line

1. **Problem Statement:** Find the length of segment $DF$ in triangle $DEF$ given:

1. დავწეროთ ამოცანა: გვაქვს წრეწირის რადიუსი $20$ სმ, ორი ურთიერთგადამკვეთი ქორდა სიგრძეებით $32$ სმ და $20\sqrt{3}$ სმ, რომლებიც ურთიერთმართობიან. უნდა ვიპოვოთ მონაკვეთები, რომლებ

1. **Problem statement:** We have a large regular pentagon with side length $x$ and diagonal length $y$. After drawing all diagonals, a smaller pentagon is formed inside. We want t

1. **State the problem:** A ladder 8 m long is leaning against a building. The bottom of the ladder is 3 m from the building. We need to find how high the ladder reaches on the bui

1. The problem is to determine if two lines \(a\) and \(b\) are coplanar, meaning they lie in the same plane. 2. Two lines are coplanar if they are either parallel, intersecting, o

1. **Problem statement:** We have a cone with apex A and a cylinder inscribed inside it. Given AG = 12 cm (height of the cone), AC = 2\sqrt{3} cm (radius of the cylinder's top circ

1. **Stating the problem:** We have a cone with vertex A and base circle with radius FG, and inside it a cylinder with radius CD. Given AG = 12 cm, AC = 2\sqrt{3} cm, and \angle AG

1. **Problem Statement:** We have a triangular plot with three teams Alpha, Beta, and Charlie located at the corners. The area of the triangle is 37 square units. The coordinates o

1. **Stating the problem:** We have a cone with vertex $A$ and base radius $FG$. Inside the cone is a cylinder with top circle radius $CD$. Given $AG=12$ cm, $AC=2\sqrt{3}$ cm, and

1. **State the problem:** We have a cone with radius $2.4$ cm and slant height $6.3$ cm, and a hemisphere with radius $R$ cm. The total surface area of the cone equals the total su

1. **Problem statement:** Given an acute angle $\widehat{xOy} > 50^\circ$, points $A$ on ray $Ox$ and $B$ on ray $Oy$ such that $OA = OB$. $H$ is the midpoint of segment $AB$. 2. *

1. **Problem statement:** Given a circle with center $O$ and diameter $BOD$, angles $\angle ABD = 30^\circ$ and $\angle AXD = 70^\circ$, find: (i) the reflex angle $\angle BOC$

1. **Problem statement:** We have a right triangle MFQ with a right angle at Q, angle M is 30°, and point R lies on segment MQ. Angle at R is 135°. Segment MF is 12. We need to fin

1. **State the problem:** We need to find the length $x$ in a right triangle where one angle is $30^\circ$ and one segment of the base is 9 cm. 2. **Identify the relevant formula:*