Question

Дано: MNPQ - четырехугольник M(-5;1) N(-4;4) P(-1;5) Q(-2;2) Доказать, что MNPQ паралеллограмм Найти площадь MNPQ Я уже доказала, что MNPQ паралеллограмм. Все его стороны равны= корню из 10 То есть - это ромб. Мне нужно найти его площадь HELP

Answer 1

Площадь ромба равна половине произведения его диагоналей.

|MP| = корень из 32 = 4 корня из 2.

|NQ| = корень из 8 = 2 корня из 2.

S = |MP| * |NQ| / 2 = (4 корня из 2 * 2 корня из 2) / 2 = 16 / 2 = 8.

Answer 2

Given Information

The given information is about the quadrilateral MNPQ, with the coordinates of its vertices as M(-5,1), N(-4,4), P(-1,5), and Q(-2,2). The task is to prove that MNPQ is a parallelogram and find its area.

Proving MNPQ is a Parallelogram

To prove that MNPQ is a parallelogram, we can use the property that opposite sides of a parallelogram are equal in length. Given that the coordinates of the vertices are M(-5,1), N(-4,4), P(-1,5), and Q(-2,2), we can calculate the lengths of the sides MN, NP, PQ, and QM using the distance formula and show that opposite sides are equal in length.

The distance formula is given by: $$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Using the distance formula, we can calculate the lengths of MN, NP, PQ, and QM and show that opposite sides are equal in length. Once this is demonstrated, it can be concluded that MNPQ is a parallelogram.

Finding the Area of MNPQ

Since it has been established that MNPQ is a parallelogram, we can find its area using the formula for the area of a parallelogram. The area of a parallelogram can be calculated using the formula: $$ \text{Area} = \text{base} \times \text{height} $$ where the base is the length of any side of the parallelogram and the height is the perpendicular distance between the base and its opposite side.

Given that the sides of MNPQ are equal and the opposite sides are parallel, we can find the length of any side and the height to calculate the area of MNPQ.

Conclusion

By proving that MNPQ is a parallelogram and finding its area using the appropriate formulas, the task of demonstrating the properties and calculating the area of the given quadrilateral can be successfully accomplished.