📐 geometry

1. **Problem statement:** Find the value of $x$ in the given triangles. 2. **Recall the triangle angle sum rule:** The sum of the interior angles in any triangle is always $180^\ci

1. **Problem statement:** We need to find the angles of a parallelogram given that the angle between two altitudes drawn from the vertex of an obtuse angle is 60°. 2. **Known facts

1. The problem asks to draw a flat shape whose area equals its perimeter. 2. Let's consider a square with side length $s$.

1. **Problem statement:** Given quadrilateral ABCD with diagonals AC and BD intersecting at P. On AC, point M is chosen such that $AP = MC$. On BD, point N is chosen such that $BP

1. The problem is to understand and apply Ceva's Theorem in geometry. 2. Ceva's Theorem states that for a triangle $ABC$, and points $D$, $E$, and $F$ lying on sides $BC$, $CA$, an

1. **State the problem:** We have a triangle with vertices C, M, N, and B, where segment MN is parallel to side CB. Given angles at M are 35° and 60°, and we need to find the angle

1. **Problem statement:** We need to find the equation of a circle with radius 6 and then find a chord passing through the intersection points of this circle and another circle. 2.

1. The problem is to draw a circle with a radius of 8 in Cartesian coordinates. 2. The standard formula for a circle centered at the origin $(0,0)$ with radius $r$ is:

1. **Problem statement:** We are given a triangle ABC with sides AB=10 cm, BC=8 cm, and CA=6 cm. We need to verify if it is a right-angled triangle and solve for its angles. 2. **C

1. The problem asks us to plot the points with coordinates (5, 3), (1, 5), (0, 5), (3, 0), and (4, 2). 2. Each point on the coordinate plane is represented as $(x, y)$, where $x$ i

1. The problem states that ABCD is a square with right angles at each vertex. 2. In a square, all sides are equal in length.

1. **State the problem:** We need to find the area of a right trapezoid with the top base $3$ units, the bottom base $2$ units, and the height $6$ units. 2. **Formula for the area

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