📐 geometry

1. Let's first state the problem: We want to determine if there exists a triangle for which a given relation holds. However, the relation itself was not provided in your message. 2

1. Bài toán yêu cầu xác định các tính chất hoặc công thức liên quan đến hình tam giác. 2. Hình tam giác là đa giác có ba cạnh và ba góc.

1. Bài toán yêu cầu tính diện tích trồng cỏ, biết một hình có hai cạnh là 3cm và 2cm, và có 4 bồn để trồng cỏ. 2. Giả sử mỗi bồn có hình chữ nhật với chiều dài 3cm và chiều rộng 2c

1. **State the problem:** We have a circular parlor with radius $r=4$ cm (distance from door to middle). A circular table in the middle occupies 10% of the parlor's area. We want t

1. **Problem statement:** Triangle ABC is isosceles with AC parallel to BD. Given angle C is 40°, find the values of angles $a$ and $b$. 2. **Key facts and formulas:**

1. The problem is to analyze the relation given by: $$2R = \frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca}$$

1. The problem involves the relation between the inradius $r$, circumradius $R$, and sides $a,b,c$ of a triangle, given by the equation: $$2R = \frac{1}{ab} + \frac{1}{bc} + \frac{

1. **Nêu bài toán:** Cho đường tròn (O) và hai dây cung AB, CD với CD = AB. Giao điểm K của các đường thẳng AB và CD nằm ngoài đường tròn. Vẽ đường tròn (O; K), đường tròn này cắt

1. **Stating the problem:** We have a trapezoid ABCD with BC = 14 cm and CD = 10 cm. Inside it, triangle BFC has an area of 50 square units. We need to find the area of the shaded

1. **Problem statement:** Solve the problem of finding the lengths of segments formed by two intersecting chords inside a circle without using the theorem "When two chords intersec

1. **Problem Statement:** Given a circle with points A, B, C on the circumference, chords XY, AP, and AQ such that X and Y are midpoints of arcs AB and AC respectively, and arcs AX

1. **Problem Statement:** Given triangles $\triangle ABC$ and $\triangle DEF$ with $AB = DE$, $AB \parallel DE$, $BC = EF$, and $BC \parallel EF$, and vertices $A, B, C$ joined to

1. **Problem Statement:** In parallelogram ABCD, points P and Q lie on diagonal BD such that $DP = BQ$. We need to prove: (i) $\triangle APD \cong \triangle CQB$

1. **Problem Statement:** Show that if the diagonals of a parallelogram are equal, then the parallelogram is a rectangle. 2. **Recall the properties:**

1. **Stating the problem:** We have a cube model with side length 7 cm. The cubes are sold in packages of 60 and 90 units. We want to find the packaging dimensions that minimize su

1. Bài toán yêu cầu chứng minh hai đường thẳng SB và SC song song với mặt phẳng (MNP). 2. Để chứng minh một đường thẳng song song với mặt phẳng, ta cần chứng minh đường thẳng đó so

1. **Stating the problem:** Calculate the area of rectangle ABCD with length BC = 14 cm and height AD = 10 cm.

1. Problem statement: Given a right triangle $\triangle ABC$ with right angle at $A$ and $AB < AC$, and altitude $AH$. Points $E$ and $F$ are the feet of perpendiculars from $H$ to

1. **State the problem:** We have a telephone wire stretched from the top of a telephone pole to a stake in the ground. The wire length is 7 metres, and the horizontal distance fro

1. **State the problem:** We have a right triangle with one leg of length 4 km, a hypotenuse of length 8 km, and the other leg length is unknown, labeled as $b$. 2. **Formula used:

1. The problem states that circle A is given by the equation $$x^2 + (y - 1)^2 = 49$$. This represents a circle centered at $(0,1)$ with radius $7$ because the general form of a ci