📐 geometry

1. **Problem Statement:** Determine which statement can be used to conclude that quadrilateral XYZW is a parallelogram. 2. **Key Concept:** A quadrilateral is a parallelogram if bo

1. **State the problem:** We need to find values of $x$ and $y$ such that quadrilateral $ABCD$ is a parallelogram. 2. **Key property:** In a parallelogram, the diagonals bisect eac

1. **Problem statement:** Given triangle ABC with M as midpoint of BC, X on segment AM, Y the foot of the altitude from X to BC, Z on segment XY, and D, E the feet of altitudes fro

1. Bài toán: Cho tam giác ABC với điểm D trên AC sao cho $AO=\frac{1}{2}AC$ và điểm E trên BD sao cho $ED=\frac{1}{3}BC$. Đường thẳng nối AE kéo dài cắt BC tại điểm M. Tính độ dài

1. **State the problem:** We are given the area of one face of a cube as 16 cm² and need to find the total surface area of the cube. 2. **Formula and rules:** The area of one face

1. The problem is to understand when to add or subtract in the Pythagorean theorem context. 2. The Pythagorean theorem states that in a right triangle with legs $a$ and $b$, and hy

1. **Problem Statement:** Given two parallel lines $a \parallel b$ and angles with measures $m\angle 2 = 2x$, $m\angle 3 = 3x - 10$, and $m\angle 4 = 5x - 10$, find the measures of

1. **Nêu bài toán:** Cho tam giác ABC vuông tại A, AH vuông góc với BC, HC > 2HB, D là điểm đối xứng của A qua B. 2. **Chứng minh góc AHD + góc ACD = 180 độ:**

1. **State the problem:** We need to find the measure of angle $x$ in a triangle with sides 40 cm, 18 cm, and 25 cm, where $x$ is the angle between the 40 cm and 25 cm sides. 2. **

1. 문제 설명: 정육면체 64개를 4x4x4 형태로 쌓은 큰 정육면체의 겉면을 모두 색칠했습니다. 이때 한 면이라도 색칠된 작은 정육면체의 개수를 구하는 문제입니다. 2. 공식 및 규칙:

1. 문제: 돌레가 98cm인 정육면체의 전개도를 접어 정육면체를 만들었습니다. 이 정육면체의 모든 모서리 길이의 합을 구하세요. 2. 정육면체의 돌레는 한 변의 길이의 12배입니다. 즉, 돌레 $D=12a$이고 여기서 $a$는 한 변의 길이입니다.

1. **Problem Statement:** We have a tessellation figure and a heptagon with interior angles expressed in terms of $x$. Also, a straight line $AOB$ with angles labeled in terms of $

1. The problem states that triangle ABC is similar to triangle XYZ, and we know the side length XZ = 10 units. 2. Given triangle ABC has sides AB = 9, BC = 12, and hypotenuse AC =

1. **Problem 1.3:** Two similar triangles have areas 18 cm² and 32 cm². The base of the smaller triangle is 6 cm. Find the base of the larger triangle. 2. **Formula and rule:** For

1. **Stating the problem:** We need to find which regular shape among Circle, Kite, Parallelogram, and Rhombus has the least number of orders of symmetry. 2. **Understanding symmet

1. **State the problem:** We have a frustum formed by removing a smaller square-based pyramid from a similar larger one. The base edge of the larger pyramid is 48 mm, and the slant

1. **Problem statement:** Given an acute, scalene triangle $\triangle ABC$ with $AB < AC$, $O$ is the circumcenter (intersection of perpendicular bisectors), $H$ is the orthocenter

1. პრობლემა: მომხმარებელი ითხოვს ხაზის დახაზვას, მაგრამ არ არის მოცემული რაიმე ინფორმაცია ხაზის შესახებ. 2. ხაზის დახაზვისთვის საჭიროა მინიმუმ ორი წერტილი ან ერთი წერტილი და დახაზვ

1. **Problem Statement:** Represent $\sqrt{9.3}$ on the number line using a geometric construction. 2. **Given:** Segment $AB = 9.3$ cm, point $C$ on $AB$ such that $AC = 1$ cm and

1. The problem is to represent $\sqrt{9.3}$ on a number line. 2. We know that $\sqrt{9.3}$ is the positive number which, when squared, equals 9.3.

1. **State the problem:** We are given a right triangle where the hypotenuse is 225 and one leg is 7. We need to find the length of the other leg. 2. **Formula used:** In a right t