📐 geometry

1. **Questão 02: Converter a equação polar da circunferência $r=6\cos(\theta)$ para a forma cartesiana.** A fórmula para converter coordenadas polares para cartesianas é:

1. **Stating the problem:** We have a cyclic quadrilateral inscribed in a circle with angles labeled as 50°, z, 96°, x, 40°, and y. We need to find the unknown angles z, x, and y.

1. The problem asks to construct geometric shapes and tilings based on Euclid's equilateral triangle construction. 2. **Constructing a regular hexagon (1.2.1):**

1. The problem states that the angles $A$, $B$, and $C$ satisfy the equation $A + B + C = \pi$. 2. We want to use this fact to express $A + B$ in terms of $C$.

1. **State the problem:** Three regular polygons A, B, and C meet at a point, and their interior angles are in the ratio $a:b:c=3:4:5$. We need to show that polygon C has twice the

1. **State the problem:** Calculate the surface area of a cylinder with radius $r=2$ cm and height $h=4$ cm. 2. **Formula for surface area of a cylinder:**

1. We are asked to find the surface area of three different cylinders. 2. The formula for the surface area $S$ of a cylinder is:

1. **Problem Statement:** Find the missing angle \(\angle C\) in a right triangle using the cosine ratio.

1. **State the problem:** We are given a triangle divided into two smaller right triangles with angles labeled as follows: one small triangle has an angle $3x - 5$, the other has a

1. Задача 10: Точките P и Q лежат на страните AC и BC на ΔABC с условия AP = \frac{1}{4}AC и CQ = \frac{1}{3}BC. Трябва да намерим вектора AB чрез векторите p = AP и q = CQ. 2. Изп

1. **Problem:** Find the area and perimeter of a rectangle with length 7 ft and width 5 ft. 2. **Formulas:**

1. **State the problem:** Prove that triangles $\triangle RTU$ and $\triangle RUS$ are similar. 2. **Given:**

1. **Problem Statement:** Identify which pairs of angles are consecutive interior angles given the parallel lines and transversal. 2. **Definition:** Consecutive interior angles ar

1. **Problem Statement:** Given two parallel lines $\overrightarrow{FH}$ and $\overrightarrow{IK}$ intersected by a transversal $\overrightarrow{EL}$ passing through points $G$ and

1. **Problem Statement:** We have a circle with two secants intersecting outside the circle at point H. One secant passes through points S and H, extending beyond H to F, with segm

1. **Problem Statement:** We have a circle with points L and G on the circumference. A chord FG passes through the circle and is intersected by line LS at point S. The chord FG is

1. **Problem 1:** Calculate the distance from the surface to the centre of the Earth given the Earth's circumference is 40000 km. 2. The formula relating circumference $C$ and radi

1. **Problem statement:** Calculate the radius or diameter of a circle given its circumference, correct to three significant figures. 2. **Formula:** The circumference $C$ of a cir

1. **Problem statement:** Find the area and perimeter of a rhombus with diagonals of lengths 30 cm and 16 cm. 2. **Formulas:**

1. **Problem statement:** We have a pyramid with rectangular base ABCD where $AB=20$ cm, $BC=16$ cm, and the vertex $E$ such that $AE=BE=CE=DE=17$ cm. We need to find the angle bet

1. **Problem statement:** Given points A(7,7), B(8,4), and C(6,0) lie on the same circle, find: a) the equation of the perpendicular bisector of AB