📐 geometry

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:*

1. 問題陳述： 我們有邊長為5公分的正三角形，將20個這樣的三角形以邊相連的方式連接成一排，求整個圖形的周長。

1. **Problem:** Calculate the perimeter when 5 chocolate bricks, each 10 cm long and 4 cm wide, are connected horizontally. 2. **Formula:** The perimeter $P$ of a rectangle is $P=2

1. نبدأ بكتابة معادلة الدائرة العامة التي تمر بالنقاط المعطاة. معادلة الدائرة في المستوى هي: $$ (x - h)^2 + (y - k)^2 = r^2 $$

1. **State the problem:** We need to find the area of the square given the information about the two triangles inside it and one side length. 2. **Analyze the given information:**

1. **Stating the problem:** We have a right triangle PQR with \(\angle Q = 90^\circ\) and \(\angle R = 2\angle P\). We want to analyze the triangle using the given angle relationsh

1. সমস্যাটি হলো: একটি ত্রিভুজ ABC যেখানে BC = 12 সেঃমিঃ। ত্রিভুজের ভিতরে M এবং N বিন্দু AC এবং BC রেখায় অবস্থিত। ত্রিভুজ CMN এর ক্ষেত্রফল 9 বর্গ সেঃমিঃ। আমাদের AM এর মান নির্ণয় ক

1. সমস্যাটি হলো: ত্রিভুজ CMN এর ক্ষেত্রফল 9 বর্গ সেমি এবং BC = 12 সেমি, AM এর মান নির্ণয় করতে হবে। 2. প্রথমে, আমরা বুঝি যে BC = 12 সেমি এবং ত্রিভুজ ABC তে B কোণটি সমকোণ।

1. Problem: Given a triangle with BC = 12 cm and the area of triangle CMN is 9 square cm, find the length of AM. 2. Formula: The area of a triangle is given by $$\text{Area} = \fra

1. **State the problem:** Find the coordinates of the image (reflection) of the point $(-4,5)$ about the line $x + 3y - 1 = 0$. 2. **Formula and explanation:** The reflection of a

1. **State the problem:** We have a rectangular paper folded along its diagonal AB. The ratio of the area of Figure 2 (shaded part) to Figure 1 (remaining part) is 4:7. The shaded

1. **Problem statement:** Calculate the surface area of a triangular prism. 2. **Formula:** The surface area $SA$ of a triangular prism is given by:

1. The question asks if point A can have the same arc as point E. 2. In geometry, an arc is a portion of the circumference of a circle defined by two points.

1. **Problem Statement:** Given a circle with center O and points A, B, C, D, E on or inside the circle, with lines AB, BC, CD, DE, AE drawn as described, and angles $x$ at vertex