📐 geometry

1. **Problem statement:** Given two adjacent angles on a straight line, where the left angle is $4d^\circ + 16^\circ$ and the right angle is $2d^\circ + 14^\circ$, find the value o

1. **Problem Statement:** We need to find the area of quadrilateral GHEN given $HG = 30\sqrt{2}$ and $EG = 70$.

1. **Problem Statement:** We have a square ABCD with diagonals AC and BD intersecting at N. Points E, F, and G lie on sides AD, AC, and CD respectively such that DEFG is a rectangl

1. Bài toán yêu cầu tính diện tích hình chữ nhật ABCD biết rằng $AI=\frac{2}{5}AB$ và diện tích tam giác BIOC là 279 cm$^2$. 2. Ta biết rằng trong hình chữ nhật, các đường chéo cắt

1. **State the problem:** We need to find the area of the shaded part, which consists of two identical parallelograms joined along their longer base inside a rectangle. 2. **Identi

1. **State the problem:** The angles in a triangle are in the ratio 3 : 1 : 2. We need to find the size of each angle. 2. **Recall the rule:** The sum of the angles in any triangle

1. The problem states that the square's jugfol is 630. We need to clarify what "jugfol" means, but assuming it refers to the area of the square. 2. The formula for the area of a sq

1. **State the problem:** We have a rectangle where the length is 7 cm longer than the width.

1. The problem involves several geometric constructions: drawing lines, dividing them proportionally, constructing angles by bisection, and constructing triangles with inscribed an

1. **Problem:** Draw a line AB 60 mm long and bisect it. **Step 1:** Draw a straight line segment AB of length 60 mm.

1. **Problem:** Draw a line segment AB 60 mm long and bisect it. 2. **Formula and rules:** To bisect a line segment means to divide it into two equal parts at its midpoint.

1. **Stating the problem:** We are given the curved surface area formula of a cone: $$A = \pi r \sqrt{h^2 + r^2}$$

1. **State the problem:** We have a right-angled triangle with hypotenuse length $b$ cm and other sides $f$ cm and $g$ cm.

1. **Problem:** Which theorem or postulate can be used to show $\triangle GBE \cong \triangle ABD$? 2. **Recall the common triangle congruence postulates:**

1. **Problem statement:** We have a board of length 14.5 cm leaning against a vertical wall 10.5 cm high. The angle between the ground and the board is $a$ (in degrees). We want to

1. **Problem statement:** Find the value of $k$ and the area of a trapezium given certain conditions (not specified in the question, so we assume $k$ is a parameter related to the

1. **State the problem:** We have a trapezium PQRS with parallel sides PQ and SR. PQ = $2k + 2$ cm, SR = $4k - 1$ cm, height QR = $3k$ cm, and area $L = 150$ cm². We need to expres

1. **Problem statement:** Given trapezium AECB with AE \parallel BC, find the angles $\angle x$, $\angle y$, and $\angle z$. 2. **Key property:** In trapezium AECB, since AE is par

1. **State the problem:** We have two similar quadrilaterals CDEF and SRUT. Given the side lengths of CDEF: $CD=30$, $DE=63$, $EF=39$, $FC=81$, and some sides of SRUT: $TU=13$, $UR

1. **Problem statement:** We have an analogue clock with an hour hand of length 5 cm and a minute hand of length 10 cm. At 6:00, after $x$ minutes ($0 < x < 60$), the hour and minu

1. **Stating the problem:** We are given a circle with points S, T, U, P, Q, R on its circumference and a point O inside the circle. We know the angle at T formed by points S, T, U