📐 geometry

1. The problem asks to find the new coordinates of triangles after applying given translations. 2. Translation means moving every point of a shape by the same amount in a given dir

1. **State the problem:** We are given a diagram with angles expressed in terms of $x$ and $y$ in degrees. We need to find the values of $x$ and $y$. 2. **Identify the angles and r

1. **הבעיה:** נתון כי הנקודה C היא אמצע הקטע AD, ושהקטעים DE ו-AB מקבילים ($DE \parallel AB$). יש להוכיח כי C היא גם אמצע הקטע EB. 2. **הגדרות וכללים חשובים:**

1. **State the problem:** We have triangle ABC with points M on AB and K on AC. Given ratios: $\frac{AM}{MB} = \frac{3}{2}$ and $\frac{AK}{KC} = \frac{2}{3}$.

1. **Stating the problem:** We have triangle ABC with points M on BC such that $BM:MC=2:3$, and point K on segment AM such that $AK:KM=2:1$. We want to find the ratios of areas $\f

1. **State the problem:** We have triangle ABC with a right angle at C, angle A = 120°, side AC = 8, and side BC = 5. We need to find the area $S_{ABC}$. 2. **Recall the formula fo

1. **Problem Statement:** Find the missing interior angle of the Hospital land area, which is a triangle with given angles 63.4° and 45°. 2. **Formula Used:** The sum of interior a

1. **Problem Statement:** We are given an isosceles triangle $\triangle ABC$ with $AB = BC$ and an equilateral triangle $\triangle DEF$ inscribed inside it such that points $D$ and

1. **Problem Statement:** In triangle $\triangle ABC$ with $AB = AC$, points $D, E, F$ lie on sides $AB, BC, CA$ respectively such that $DE = EF = FD$, forming an equilateral trian

1. **State the problem:** We are given a circle with radius $r = 3$ meters and an angle $\theta = \frac{5\pi}{12}$ radians. We need to find the arc length corresponding to this ang

1. The problem asks to draw the plan view of a 3D shape given its front elevation and side elevation. 2. The front elevation shows a red right triangle on a square grid.

1. **Problem statement:** Given circle (O) and point A outside (O), with tangents AB and AC to (O) at points B and C respectively. A secant ADE is drawn such that ray AD lies betwe

1. **State the problem:** We have a triangle ABC with angles $a$, $b$, and $c$. Given: - $a$ is 13 less than $c$, so $a = c - 13$

1. **Problem statement:** Given a right isosceles triangle ABE with hypotenuse AE = 10 cm, find the area of the square whose side is the leg BE of the triangle. 2. **Formula and ru

1. **Problem statement:** We have points A, B, C, D, and E on a circle with given angles 26°, 52°, and 125°, and unknown angles $x$ and $y$. We need to find $x$ and $y$. 2. **Key r

1. **State the problem:** We have three identical rectangles arranged in an L-shape on a coordinate plane. The bottom-left corner of the lower rectangle is at $(2,3)$ and the top-r

1. **State the problem:** We are given a quadrilateral with three angles labeled as 110°, 110°, and 70°, and we need to determine if this quadrilateral is a parallelogram. 2. **Rec

1. **State the problem:** We are given expressions for segments SU and SW in terms of $b$ and need to find the value of $b$ that makes quadrilateral TUVW a parallelogram. 2. **Reca

1. **Problem statement:** Given a regular quadrilateral pyramid $S.ABCD$ with base side length $a$ and center $O$. Points $M$ and $N$ are midpoints of edges $SA$ and $BC$ respectiv

1. **State the problem:** We need to find the areas of three parts: (a) the rectangle, (b) the triangle, and (c) the whole composite shape.

1. **State the problem:** We are given the circle equation $$4x^2 + 4y^2 - 8x - 12y + 1 = 0$$ and a point $$P(3, 2.5)$$ outside the circle. We need to find the length of the tangen