📐 geometry

1. **Problem statement:** We have a right triangle ABC with a right angle at B. The side AB is 12 cm, and the hypotenuse AC is 13 cm. We need to find the length of side BC. 2. **Fo

1. **Problem Statement:** Find the height and diameter of an oblique circular cylinder given the axis length and the angle with the base plane.

1. **Nêu bài toán:** Cho tam giác đều ABC có cạnh bằng 3. Điểm M thỏa mãn: $$2MA^2 + MB^2 + MD^2 = 18$$. Tìm bán kính đường tròn mà tập hợp điểm M thuộc về. 2. **Công thức và kiến

1. **Problem Statement:** Label an oblique circular cylinder with a 45 degrees angle with the plane of its base.

1. **Stating the problem:** We have a half-cylinder shape with a bottom diameter of 64 cm, a top diameter of 110 cm, and a total length of 455 cm. It is cut into 4 equal pieces alo

1. **Stating the problem:** We have three angles related by the diagram: one angle is $5y$, another is $4y$, and the angle between these two rays is $y$. We want to find the value

1. **Problem statement:** Find the coordinates of the point that divides the points $P(8,9)$ and $Q(-7,4)$ internally in the ratio $2:3$ and externally in the ratio $4:3$. 2. **For

1. Vous avez demandé de reproduire une figure, mais aucun détail ou description de la figure n'a été fourni. 2. Pour pouvoir vous aider, j'ai besoin d'une description précise ou d'

1. **Énoncé du problème :** Vérifier que $\overrightarrow{A J} = 3 \overrightarrow{A C}$ sachant que $\overrightarrow{J C} = \frac{2}{3} \overrightarrow{J A}$.

1. **Problem statement:** You asked how we found that $\angle BDA = 40^\circ$ given $\angle DBC = 40^\circ$ in the circle. 2. **Key property used:** Angles subtended by the same ch

1. **Problem statement:** We are given a circle ABCD with center O, where AB = AC and \(\angle DBC = 40^\circ\). We need to find \(\angle ABD\). 2. **Given information and properti

1. The problem asks which of the given points lies inside the shaded triangular region in the fourth quadrant. 2. From the description, the shaded region is bounded above by a line

1. **Problem:** Find the values of $x$, $y$, and $z$ in the given circles with center $O$ where angles around the center sum to $360^\circ$. 2. **Formula:** The sum of angles aroun

1. **Stating the problem:** We have a triangle with points A, L, C, T, B, D where segments AL, LC, and CT are equal in length. The angle \(\angle TCD = 96^\circ\) is given, and we

1. **Problem Statement:** We need to analyze a triangle with an exterior angle formed by extending one side, a line inside the triangle creating two interior angles, and at least f

1. **Problem:** Find the coordinates of point P which is on the x-axis and at a distance of $\sqrt{10}$ units from the line $3x - y + 2 = 0$. 2. **Formula for distance from a point

1. **Problem:** Find the coordinates of point P which is on the x-axis and at a distance of $\sqrt{10}$ units from the line $3x - y + 2 = 0$. 2. **Formula:** The distance $d$ from

1. **State the problem:** We need to find the area of the shaded right triangle. The two triangles share a vertical side of length 5 meters, and the base of the right triangle is 5

1. **State the problem:** Find the coordinates of the centroid of triangle $\triangle PQR$ with vertices $P(-7,7)$, $Q(4,2)$, and $R(3,8)$.\n\n2. **Formula for centroid:** The cent

1. **Problem statement:** We have a large rectangle divided into smaller rectangles with given areas: 30, 21, 10, 40, 12, and 90 cm². All side lengths are whole numbers. We need to

1. **Problem Statement:** We have a triangle with one angle measuring 87° and two sides labeled as 19a and 12a. We need to find the value of the variable $a$ representing an angle