📐 geometry

1. **Problem statement:** Given points $A(3,0,0)$, $B(1,2,-3)$, and $C(0,1,-1)$, find the coordinates of point $D$ such that $ABCD$ is a parallelogram, and then find the area of pa

1. **بيان المسألة:** نريد إيجاد طول المسار الذي يمر بالنقاط A - B - F - E + B = A، معطى أن (AB) || (DF)، وDE = 4.5 km، CP = 5 km، AC = 7.5 km.

1. **Problem Statement:** We are given a right triangle with vertices A, B, and C, where the right angle is at vertex C. The side opposite vertex A is 5 units, the side opposite ve

1. **Problem Statement:** We are given a right triangle with vertices B, C, and A. The right angle is at vertex C. Side BC measures 6, side BA measures 7, and side CA (opposite ang

1. **Problem Statement:** We are given a right triangle with vertices B, C, and A. Side BC is vertical with length 2 units, side BA is the hypotenuse with length 5 units, and angle

1. **Problem statement:** A parallelogram has the largest side $a=55$ cm. A diagonal makes angles $30^\circ$ and $50^\circ$ with the pair of adjacent sides. We need to find the len

1. **Problem statement:** The university owns a triangular piece of land with two sides measuring 650 m and 720 m, and the angle between them is 75 degrees. We need to find:

1. **State the problem:** We have a right triangle ABC with a right angle at C. Side AC is 5 units, and angle A is 40°. We want to find the length of side AB. 2. **Recall the relev

1. **State the problem:** We have a right triangle with vertices A, B, and C, where the right angle is at C. The side opposite angle C (AC) is 6 units, the side opposite angle A (A

1. The problem involves finding the measure of angle $\angle 1$ in a right triangle where one angle is $90^\circ$ and another is $40^\circ$. 2. Recall that the sum of angles in any

1. **Problem Statement:** We are given a triangle with angles labeled 5, 6, and 7 at the top vertex, and angles 1, 2 at the bottom left vertex, and angles 3, 1 at the bottom right

1. مسئله: یافتن معادله یک دایره با مرکز و شعاع مشخص است. 2. فرمول دایره: معادله دایره به شکل $$ (x - h)^2 + (y - k)^2 = r^2 $$ است که در آن $ (h, k) $ مرکز دایره و $ r $ شعاع آن اس

1. **Problem statement:** We have a pentagon with given angles 30°, 40°, and 90°, and we need to find the angle $x$ at the bottom-left corner. 2. **Key fact:** The sum of interior

1. The problem involves identifying the pattern of shaded triangles in a sequence of four small squares, each divided into a 4x4 grid with diagonal lines. 2. Each small square has

1. The problem is to analyze a right triangle on the Cartesian plane. 2. In a right triangle, one angle is $90^\circ$.

1. **Problem Statement:** Given that $AF \parallel HK$ and the angles at $H$ are $40^\circ$, $30^\circ$, and $30^\circ$, find the angles $\angle ABC$, $\angle ABH$, $\angle IHJ$, $

1. **Problem Statement:** Given parallelogram ABCD with diagonal BD, points M and N lie on BD such that segment MN is parallel to BD. We need to prove that the area of triangle BAM

1. **Problem Statement:** Find the coordinates of the limiting points of the coaxial system of circles given by the equations: $$x^2 + y^2 = 2x + 8y + 11$$

1. **State the problem:** We have a triangle and its rotated image on the coordinate plane. We need to find: a) The coordinates of the centre of rotation.

1. The problem is to find the distance given by the formula $$\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ which calculates the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$

1. The problem is not fully clear, but it seems to involve a square area of 900 m² and some measurements: 1 m, 5 m, speeds of \(\sqrt{2}\) km/h and 10 km/h, and an angle \(\theta\)