📐 geometry

1. **Problem statement:** Verify that the quadrilateral formed by joining the midpoints of the sides of quadrilateral PQRS with vertices P(0, 6), Q(-6, -2), R(2, -4), and S(4, 2) i

1. **Problem:** Verify that the diagonals of quadrilateral ABCD with vertices A(0,0), B(2,3), C(5,1), and D(3,-2) are perpendicular. 2. **Formula and rules:**

1. The problem asks for the length of the hypotenuse of a right triangle with legs measuring 11 cm and 10 cm. 2. We use the Pythagorean theorem, which states that in a right triang

1. The problem asks for the simplest way to find angle $Y$ in triangle $XYZ$ where $\angle X = 40^\circ$, $\angle Z = 60^\circ$, and side $XZ = 12.0$ cm. 2. Recall that the sum of

1. **Stating the problem:** We are given a triangle with angles $43^\circ$, $100^\circ$, and $37^\circ$ at vertices A, B, and C respectively, and side lengths $AB=3$ cm, $AC=7$ cm,

1. The problem is to understand the definition and relationship between two lines in geometry. 2. Two lines can be related in several ways: they can be parallel, intersecting, or c

1. **Problem statement:** Find the area of one small shaded triangle inside a large equilateral triangle with side length 12, given all small triangles are equal in area. 2. **Form

1. The problem asks to find the length of the hypotenuse for right triangles given the lengths of the legs. 2. We use the Pythagorean theorem formula for right triangles:

1. **Problem statement:** We have a square with a diagonal length of $8\sqrt{2}$ cm. We need to find the area of the shaded triangle, which is one of the four triangles formed by t

1. The problem states that six identical right-angled triangles are arranged to form a rectangle. 2. The height of the rectangle is given as 7 cm.

1. **Problem statement:** Given that ABC and CDE are straight lines and AE is parallel to BD, find the sizes of angles: a) $\angle ABD$

1. **Stating the problem:** We are given a triangle ABC with points D, E, F, G, H, and I extending from it. We know angles AFC = $x^\circ$, ACB = $y^\circ$, BCF = $40^\circ$, and A

1. **Stating the problem:** We have points $l$, $m$, and $n$ on a circle with center $O$. The angle $\angle N M$ (interpreted as $\angle N M O$ or $\angle N M L$ depending on conte

1. **Problem Statement:** Find the perimeter and area of the given shape, which is a right triangle with a smaller square inside it at the bottom-left corner. 2. **Understanding th

Let's learn about circles! 🎉 1. Imagine a circle with a center at point $(-5,0)$ and radius $2.3$ units.

Let's learn about a circle! 🎉 1️⃣ Imagine the center of the circle is a balloon 🎈 at point (0,3).

1. Given: $AB=3$, $BC=\sqrt{3}$, $AC=b$, and $\angle ACD=150^\circ$. We need to find the value of $4b^2 - 6\sqrt{3}$. 2. Since $A$, $B$, and $C$ form a triangle, we can use the Law

1. Bài toán yêu cầu chứng minh các tính chất hình học trong tam giác và các điểm liên quan. 2. Ta bắt đầu với câu 3A, phần a): Cho tam giác ABC, I là trung điểm BC, D thuộc tia đối

1. **Problem statement:** Calculate the area of quadrilateral ABCD where angles at vertices A and C are right angles, and the sides are given as AD = 56 cm, AB = 33 cm, BC = 16 cm,

1. The problem is to find the formula for the number of interior angles of a polygon. 2. A polygon with $n$ sides has $n$ interior angles.

1. **State the problem:** We are given two triangles \(\triangle ABC\) and \(\triangle XYZ\) with some side lengths and angle information. We need to find missing side lengths, ide