Question

use the dropdown menus and answer blanks below to prove the quadrilateral is a rhombus.

answer
attempt 1 out of 2

i will prove that quadrilateral mnop is a rhombus by demonstrating that

Explanation:

First, identify the coordinates of all vertices from the graph: M(-5, -8), N(2, -2), O(3, 7), P(-4, 1). A rhombus can be proven by showing all four sides are of equal length, using the distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ for each side.

1. Calculate length of MN:

$\sqrt{(2-(-5))^2+(-2-(-8))^2}=\sqrt{7^2+6^2}=\sqrt{49+36}=\sqrt{85}$

1. Calculate length of NO:

$\sqrt{(3-2)^2+(7-(-2))^2}=\sqrt{1^2+9^2}=\sqrt{1+81}=\sqrt{85}$

1. Calculate length of OP:

$\sqrt{(-4-3)^2+(1-7)^2}=\sqrt{(-7)^2+(-6)^2}=\sqrt{49+36}=\sqrt{85}$

1. Calculate length of PM:

$\sqrt{(-5-(-4))^2+(-8-1)^2}=\sqrt{(-1)^2+(-9)^2}=\sqrt{1+81}=\sqrt{85}$

All four sides have equal length, which satisfies the definition of a rhombus.

Answer:

all four sides of quadrilateral MNOP are congruent (equal in length)