📐 geometry

1. **Problem statement:** Draw the image of triangle A after (i) a reflection in the line $y = -x$ and (ii) a translation by the vector $(-2, -9)$.

1. The problem is to find the area of a triangle. 2. The formula for the area of a triangle is given by:

1. **Problem 1: Rectangle ABCD with diagonal angle and side length** Given: Rectangle ABCD with angle between diagonals $54^\circ$, side $CD = 19$ cm, find $x = AD$.

1. Given an inscribed circle with chords EF, FH, HG, and GE intersecting at O, and the angle \(\angle EOH = 53^\circ\), find \(\angle EFHI\). 2. The problem states that \(\angle EO

1. **Problem Statement:** Given a circle with points E, F, G, H on the circumference and center O, inside the circle there is a triangle with an angle marked 53° at point G. We nee

1. **State the problem:** We need to find the coordinates of a point whose abscissa (x-coordinate) is $-4$ and which is at a distance of 15 units from the point $(5,-9)$. 2. **Form

1. The problem asks to find the coordinates of a point whose abscissa (x-coordinate) is -4. 2. The abscissa is the x-value of a point in the Cartesian coordinate system. The ordina

1. **Problem statement:** Find the equation of a circle passing through points $(1,2)$ and $(3,4)$ and tangent to the line $3x + y - 3 = 0$. 2. **General form of a circle:** The eq

1. **Problem statement:** Find the equation of the circle passing through points $(1,2)$ and $(3,4)$ and tangent to the line $3x + y - 3 = 0$. 2. **General form of a circle:** The

1. **Problem statement:** Find the shortest distance between the two parallel lines \(l_1: y = 2x + 4\) and \(l_2: 6x - 3y - 9 = 0\). 2. **Rewrite lines in standard form:**

1. **Problem statement:** We have a right triangle with hypotenuse $a$, opposite side to the 30° angle is 7, and adjacent side $b$ to be found. 2. **Formula and rules:** In a right

1. **Problem Statement:** We have a right triangle with a 30° angle. The side adjacent to the 30° angle is 12, the hypotenuse is $b$, and the side opposite the 30° angle is $a$. We

1. **Problem statement:** We have a right triangle with one leg of length 7, an angle of 45° adjacent to the unknown side $a$, and the hypotenuse $b$. We need to find the length of

1. **State the problem:** We have a right triangle with one leg of length 7, an angle of 45° opposite side $a$, and hypotenuse $b$. We need to find the length of side $b$. 2. **Rec

1. **Problem Statement:** We are given a right triangle with one leg of length $x$, another leg of length 15, and the hypotenuse of length 17. We need to find the value of $x$. 2.

1. **State the problem:** We need to find the value of $x$ in a right triangle where the sides are 17, 20, and $x$, with the right angle between the sides 17 and $x$. 2. **Formula

1. **State the problem:** We need to find the length of the hypotenuse $x$ in a right triangle where the legs are 15 and 11. 2. **Formula used:** According to the Pythagorean theor

1. **State the problem:** We need to find the length of the hypotenuse $x$ in a right triangle where the legs are 6 and 10. 2. **Formula used:** According to the Pythagorean theore

1. **State the problem:** We need to solve for $b$ in a right triangle where one leg is 6, the other leg is $b$, and the hypotenuse is 11. 2. **Formula used:** The Pythagorean Theo

1. **Stating the problem:** We are given a right trapezoidal prism with dimensions 8 cm (length), 4 cm (height of trapezoid), and 5 cm (base of trapezoid). We need to find the volu

1. **Problem statement:** Calculate the total volume of a triangular prism with a base triangle height of 35 cm and prism length of 120 cm. 2. **Formula:** The volume $V$ of a pris