📐 geometry

1. Muammo: To‘rtburchak A(2; 2\sqrt{3}), B(5; 5\sqrt{3}), C(9; 3\sqrt{3}), D(3; \sqrt{3}) nuqtalarida berilgan. \n\n a) \sin \beta ni toping, bu yerda \beta = \angle BAD. \n\n b) A

1. **Stating the problem:** POR and QOR form a linear pair, meaning their angles add up to 180 degrees. Given that $a - b = 80$, find the values of $a$ and $b$. 2. **Formula and ru

1. Problem statement: Given a quadrilateral ABCD with vertices A(2, 2\sqrt{3}), B(5, 5\sqrt{3}), C(9, 3\sqrt{3}), and D(3, \sqrt{3}), find (a) \sin \beta where \beta = \angle BAD,

1. **State the problem:** We have an isosceles triangle ABC with AB = AC = 13 cm and BC = 10 cm. Points D and E lie on AC and AB respectively such that AD = AE = 6.5 cm, and ED is

1. **State the problem:** We want to find the largest area of a rectangle inscribed in a semicircle of radius 2. The rectangle's base lies along the diameter from $-x$ to $x$, and

1. **Problem statement:** A ship sails 20 km east from port P, then 25 km south, then 30 km east to reach point Q. 2. **Find the total distance sailed from P to Q:**

1. **Problem statement:** Find the angle between the line segments AB and AC where \(A=(2,3)\), \(B=(-1,-4)\), and \(C=(0,-2)\). 2. **Formula used:** The angle \(\theta\) between t

1. **Problem statement:** Given a circle with center $O$ and diameter $QS$, and points $P$, $R$, $S$, and $T$ as described, with $PR = PS$, prove that $$2TO \cdot QR = QS \cdot QT.

1. The problem is to understand how to draw internal and external tangents to two circles. 2. First, let's define the terms:

1. **Problem:** In triangle ABC, given that \(\angle A = \angle B = 62^\circ\), find \(\angle C\). 2. **Formula:** The sum of the interior angles of any triangle is always \(180^\c

1. **State the problem:** We need to find the length of the hypotenuse $F$ of a right triangle with legs measuring 34 and 36 units. 2. **Formula used:** According to the Pythagorea

1. **State the problem:** We have an isosceles triangle ABC with sides BA = BC, angle BAC = 70°, and a circle inscribed inside the triangle with radius 6 cm. The length BO = 17.54

1. **State the problem:** We are given two right triangles with legs labeled as follows: the first triangle has legs $2x - 4$ and $39$, and the second smaller triangle has legs $x

1. **Find the distance between the points (-2, 0) and (4, 5).** The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

1. The first problem describes a right rectangular parallelepiped with height 6 units and vertex $D_1(4,5,0)$. We need to find the coordinates of the other vertices. 2. Since $D_1$

1. The problem asks: Which of the following is a regular polygon? 2. A regular polygon is defined as a polygon with all sides equal in length and all interior angles equal.

1. **Stating the problem:** We are given two quadrilaterals: one is a square and the other is a rectangle. We need to determine which statement about these two figures is true. 2.

1. The problem asks which figures have perpendicular sides. 2. Perpendicular sides meet at a right angle (90 degrees).

1. The problem asks which figures have perpendicular sides. 2. Perpendicular sides meet at a right angle (90 degrees).

1. **Problem statement:** We are given a triangle with sides 102 cm, 91.8 cm, and 100.2 cm, and we need to find the measure of the angle $\theta$ opposite the side of length 100.2

1. **State the problem:** We are given a triangle with angles expressed as algebraic expressions: $5x - 11$, $2x + 20$, and $11x - 63$. We need to solve for $x$ and then find the m