📐 geometry

1. مسئله: نسبت طول‌های $\frac{DF}{FE}$ را در مثلث داده شده پیدا کنیم. 2. اطلاعات داده شده:

1. مسئله را بیان می‌کنیم: در مثلث ABC، داده شده است که $AD=25$ و $BD=AE=3EC$. باید نسبت $\frac{DK}{EF}$ را پیدا کنیم. 2. ابتدا باید روابط بین طول‌ها را بررسی کنیم. چون $BD=AE=3EC$،

1. **بیان مسئله:** در مثلث ABC نقاط D و E روی اضلاع AB و AC قرار دارند. خطوط عمود از D و E به BC رسم شده‌اند که به ترتیب در نقاط K و F بر BC برخورد می‌کنند. داده‌ها: $AD=3$, $BD=5$

1. **Problem Statement:** Find the value of the sum $\angle a + \angle b + \angle c + \angle d + \angle e + \angle f + \angle g + \angle h + \angle i + \angle j$ given the angles a

1. **State the problem:** Find the volume of a cone with height $h=10$ and radius $r=7$. 2. **Formula:** The volume $V$ of a cone is given by the formula:

1. **Problem Statement:** Find the area of the composite shape consisting of a rectangle and a trapezoid on top. 2. **Given dimensions:**

1. Let's first understand the problem: You are asking how a smaller line segment CJ can have a greater value than a bigger line segment DC. 2. In geometry, the "value" of a line se

1. **Problem Statement:** Given the expressions for segments:

1. **Problem statement:** We have an equilateral triangle with side length 1, and 10 points are selected inside it. We need to prove that there exist at least two points whose dist

1. **Problem Statement:** Find the measure of angle $z$ formed between a side of a regular dodecagon and a tangent line extended from a square attached to one side of the dodecagon

1. **Problem Statement:** Find the measure of angle $z$ formed between one side of a small square and an adjacent side of a regular 12-sided polygon (dodecagon). 2. **Key Informati

1. **Problem Statement:** We are given a regular polygon with an exterior angle measuring 60 degrees. We need to find the number of sides of this polygon. 2. **Formula Used:** The

1. **State the problem:** We need to find the measure of each interior angle of a regular octagon. 2. **Formula for the sum of interior angles:** The sum of interior angles of a po

1. The problem asks for the measure of one exterior angle of a regular octagon. 2. A regular octagon has 8 equal sides and 8 equal exterior angles.

1. The problem asks for the best description of a polygon that is a hexagon with six sides, where all sides are equal in length and all interior angles are equal. 2. A polygon with

1. **Problem statement:** We have a right triangle IJK with a right angle at J, angle I measuring 22°, side IJ = 16 units, and side JK = x units. We need to find the length of side

1. **State the problem:** We have a right triangle QRS with right angle at R, side QR = 80, angle RQS = 66°, and we need to find side RS = x. 2. **Identify the sides relative to an

1. **Problem 1: Total Surface Area of the Bird-bath** The bird-bath consists of a cylinder with a hemispherical depression at one end.

1. **Problem Statement:** Given a cyclic quadrilateral ABCD with diagonals AC and BD intersecting at P, and AB = DC, prove: a) $\frac{AP}{AB} = \frac{PD}{DC}$

1. **State the problem:** Given that $\overline{EB} \cong \overline{EC}$ and $\triangle AED$ is equilateral and equiangular, prove that $\triangle ACD \cong \triangle DBA$. 2. **Un

1. **State the problem:** We have triangle ABC with angles \(m\angle ACB = 3x - 14\) degrees, \(m\angle ABC = 3x + 9\) degrees, and an exterior angle \(m\angle CAD = 103\) degrees