📐 geometry

1. Problem 11: In triangle ABC, AB = AC, and points X and Y lie on AB and AC respectively such that AX = AY. Prove that \(\triangle ABY \cong \triangle ACX\). 2. Formula and rules:

1. **Problem 5:** Given in triangles $\triangle ABC$ and $\triangle PQR$, the sides satisfy $AB = QR$, $BC = PR$, and $CA = PQ$. We need to find which triangle congruence statement

1. **Problem statement:** A farmer has tethered his cow with a rope 9 metres long. We need to find the maximum grazing area available to the cow. 2. **Formula used:** The maximum g

1. **Find the size of an interior angle of a regular pentagon.** A regular pentagon has 5 equal sides and 5 equal interior angles.

1. **State the problem:** We have an acute triangle $\triangle XYZ$ with angles $\angle X = 50^\circ$ and $\angle Y = 60^\circ$. The altitudes from vertices $X$ and $Y$ intersect a

1. **Problem statement:** In triangle $\triangle ABC$, medians $AD$ and $BE$ intersect at centroid $G$. Given $AB=10$, $AC=14$, and median $AD=8$, find the length of median $BE$. 2

1. **Problem Statement:** In triangle $\triangle ABC$, the angle bisectors of $\angle A$ and $\angle B$ intersect at point $I$ inside the triangle. We need to determine which state

1. **Problem statement:** We are given a triangle \(\triangle PQR\) with point \(O\) as the intersection of the perpendicular bisectors of its sides. It is given that \(OP = OQ = O

1. **Problem statement:** In triangle $\triangle ABC$, medians $AD$, $BE$, and $CF$ intersect at point $G$. Given that $AG = 6$, find the length of $GD$. 2. **Key property:** The m

1. **Problem Statement:** Given a quadrilateral with points A, B, C, D where sides AB = BC = CD, find the measure of angle $\angle ABC$. 2. **Key Information:** Since AB = BC = CD,

1. 문제: 두 점 A(-1, a)와 B(a, 5) 사이의 선분 AB의 길이가 최소가 되도록 하는 a의 값을 구하시오. 2. 선분 AB의 길이 공식은 두 점 $(x_1, y_1)$과 $(x_2, y_2)$ 사이의 거리로,

1. The problem is to understand the different types of quadrilaterals. 2. A quadrilateral is a polygon with four sides and four angles.

Let's learn about the number $\pi$! 🎉 1. Imagine you have a big round pizza 🍕. The number $\pi$ helps us understand things about circles, like this pizza.

1. The problem states: E is a set of quadrilaterals. 2. A quadrilateral is a polygon with four sides and four vertices.

1. 문제: 두 점 A(-1, a)와 B(a, 5) 사이의 선분 AB의 길이가 최소가 되도록 하는 a의 값을 구하시오. 2. 선분 AB의 길이 공식은 두 점 사이의 거리 공식으로 주어집니다:

1. **State the problem:** We are given a rectangle ABCD with points A(3,4) and B(5,4). One of the diameters of the circle circumscribing this rectangle is given by the line $4y = x

1. **Problem Statement:** Find the total surface area of a square-based pyramid with vertical height and slant height in the ratio 4:5 and volume 384 cm³.

1. **Stating the problem:** We are given a geometric figure with angles $x$, $y$, and $z$ and the equation $x + y = 63^\circ$. We need to find the value of angle $z$. 2. **Understa

1. مسئله: در مثلثی با اضلاع ۷، ۵ و ۳ واحد، دایرهٔ محاطی خارجی بر ضلع متوسط (که طول آن ۵ است) و امتداد دو ضلع دیگر مماس است. باید نسبت تقسیم نقطه تماس بر ضلع متوسط را بیابیم. 2. ابت

1. مسئله: در یک مثلث، ارتفاع‌ها به ترتیب 4، 3 و 2 واحد هستند. باید محیط دایره محاطی داخلی مثلث را پیدا کنیم. 2. فرمول‌ها و نکات مهم:

1. The problem is to simplify the expression $n(n-3)$ and understand the formula for the sum of interior angles of a polygon, $S = (n-2)180^\circ$. 2. First, simplify the expressio