Question

Quadrilateral JKLM is graphed with vertices at J(-2,2), K(-1,-5), L(4,0), and M(3,7).

A) Prove that the diagonals JL and MK have the same midpoint.
B) Prove that segments JL and MK are perpendicular.​

Answer 1

Question (a)

Midpoint of a line segment:

`M=\left((x_1+x_2)/(2),(y_1+y_2)/(2)\right)`

Given:

• J = (-2, 2)
• L = (4, 0)

`\implies \textsf{midpoint of }JL=\left((-2+4)/(2),(2+0)/(2)\right)=(1,1)`

Given:

• M = (3, 7)
• K = (-1, -5)

`\implies \textsf{midpoint of }MK=\left((3-1)/(2),(7-5)/(2)\right)=(1, 1)`

Question (b)

Find slopes (gradients) of JL and MK then compare. If the product of the slopes of JL and MK equal -1, then JL and MK are perpendicular.

`\textsf{slope }m=(y_2-y_1)/(x_2-x_1)`

Given:

• J = (-2, 2)
• L = (4, 0)

`\implies \textsf{slope of }JL=(0-2)/(4+2)=-\frac13`

Given:

• M = (3, 7)
• K = (-1, -5)

`\implies \textsf{slope of }MK=(-5-7)/(-1-3)=3`

`\textsf{slope of }JL * \textsf{slope of }MK=-\frac13 *3=-1`

Hence segments JL and MK are perpendicular