📐 geometry

1. **Problem statement:** We have an isosceles triangle with two equal sides of length $n$ and a vertical side of length 4.2 mm. One of the base angles is $52^\circ$. We need to fi

1. The problem asks to find the value of angle $x$ in the first figure (top-left) where two parallel horizontal lines are intersected by a diagonal transversal. 2. The given angle

1. **Problem statement:** Calculate the volume and surface area of a hemispherical water tank with radius $r=3$ meters. 2. **Formulas:**

1. **Problem statement:** We need to find the area of a regular octagon with each side length $s = 10$ cm. 2. **Formula and approach:** A regular octagon can be divided into 8 equa

1. **State the problem:** Calculate the total surface area of a roof shaped as a rectangular prism with dimensions length = 15 m and width = 10 m, plus two trapezoidal sections at

1. **Problem statement:** We need to find the area of a square with side length 5 cm and then find the dimensions of a triangle with the same area and one angle of 40°. 2. **Area o

1. The problem is to understand and solve a geometry question, but since no specific problem was given, let's consider a common geometry problem: finding the area of a triangle. 2.

1. **Problem statement:** (a)(i) Reflect triangle A in the line $y = -x$.

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