📐 geometry

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 \(\theta\)

1. **Problem Statement:** Given two parallel lines $m \parallel n$ and a transversal $t$, find the value of $x$ given the angles $(x - 30)^\circ$ and $(3x - 10)^\circ$ at the inter

1. **Problem statement:** We are given a square ABCD with sides labeled as follows: AB = 18, BC = 3(y - 1), CD = 9x, and AD = 2y + 4. We need to find the values of $x$ and $y$. 2.

1. **State the problem:** We have a parallelogram ABCD with sides labeled as follows: - Side CB = $2y - 4$

1. **Énoncé du problème :** Soit ABCD un rectangle avec AB = 5 cm et BC = 2 cm.

1. **Énoncé du problème :** Soit ABCD un rectangle avec $AB=5$ cm et $BC=2$ cm.

1. **Problem Statement:** Calculate the surface area of a right triangular prism with legs of the right triangle base measuring $1.7$ feet and $2$ feet, and the prism length is $4$

1. **Problem Statement:** Calculate the volume of a right triangular prism with legs of the right triangle base measuring $1.7$ feet and $2$ feet, and the prism length is $4$ feet.

1. **Problem statement:** A cruise ship sails north for 50 km, then turns east and sails another 50 km. We need to find the exact direct distance from the start to the finish of th

1. **State the problem:** We have a right triangle with vertices M, C, and D, right angle at C. Point B lies on segment CD, and we want to find the length of BC. 2. **Given data:**

1. **Problem statement:** We are given a geometric figure with two parallel horizontal lines and three angles: 100° between the vertical line and the top horizontal line, 45° betwe

1. **Problem Statement:** We have a triangle ABC inscribed in a circle. Points P, Q, and R lie on arcs AB, BC, and AC respectively. We need to prove that $$\angle ARC + \angle CQB

1. **Problem Statement:** We have a triangle ABC inscribed in a circle. Points P, Q, and R lie on arcs AB, BC, and AC respectively. We need to prove that $$\angle ARC + \angle CQB

1. The problem asks to name the adjacent interior angles of the given triangle. 2. Adjacent interior angles in a triangle are pairs of angles that share a common side and vertex in

1. The first question asks: How many sets of exterior angles does a given triangle have? 2. A triangle has exactly 3 exterior angles, one at each vertex, formed by extending one si

1. مسئله: دو مثلث متشابه داریم که نسبت تشابه آنها ۲ است و محیط مثلث کوچکتر ۸ است. اضلاع مثلث بزرگتر به صورت $1$, $2x^2 - 3x$, $x + 3x^2$, و $4 - x$ داده شده‌اند. باید مقدار $x$ را

1. The problem is to find the formula for the perimeter of a shape. 2. The perimeter is the total distance around the boundary of a two-dimensional shape.