📐 geometry

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

1. **Problem statement:** We have a square ABCD with side length 9. A circle centered at A with radius AD is drawn, and another circle with diameter DC is drawn. These two circles

1. **Problem statement:** Given an equilateral triangle ABC with side length $2\sqrt{a}$, a point P inside the triangle satisfies $PA=2$, $PB=2\sqrt{3}$, and $PC=4$. We need to fin

1. **Problem Statement:** Given an equilateral triangle $ABC$ with side length $2\sqrt{a}$, a point $P$ inside the triangle satisfies $PA=2$, $PB=2\sqrt{3}$, and $PC=4$. We need to

1. **Problem Statement:** We have a triangle ABD with a smaller equilateral triangle DEC inside it. We know angle $\angle A = 80^\circ$ and need to find the size of angle $\angle A

1. **Problem statement:** Given a circle with chord $AB$ of length 16 units, $D$ is the midpoint of chord $AB$, and $C$ is the midpoint of arc $ACB$. The length $CD$ is 4 units. We

1. **Stating the problem:** We have a quadrilateral C D E G formed by a parallelogram C D F G and a triangle D F E. (a) We need to identify the type of quadrilateral C D E G.

1. The problem asks to write the coordinates of the vertices of the given figures. 2. For Triangle ABC, the vertices are given as:

1. The problem states that the area of a rectangle is 24 square meters. 2. The formula for the area of a rectangle is $$\text{Area} = \text{length} \times \text{width}$$.

1. **Problem statement:** We have a square of side length 15 cm with 4 identical quarter circles drawn inside it, one in each corner. The shaded part is the region inside the squar