Big Ideas Math Integrated I, 2016
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Big Ideas Math Integrated I, 2016 View details
8. Coordinate Proofs
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Exercise 15 Page 643

What are the vertical and horizontal positions of N?

Unlabeled vertex: N(h,k)
ON=sqrt(h^2+k^2)
MN=sqrt(h^2+k^2)

Practice makes perfect

To find the lengths of ON and MN, we have to know the coordinates of N. Because ND is perpendicular with the base of the triangle which runs along the x-axis, N and D are on a vertical line and must have the same x-coordinate, h. We also see that N reaches a height of k units on the y-axis, so N has the y-coordinate k.

When we know the coordinates of N, we can use the Distance Formula to find the length of ON and MN.
d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)
d_(ON) = sqrt(( h - 0)^2 + ( k - 0)^2)
d_(ON) = sqrt(h^2 + k^2)
We will also calculate the length of MN.
d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)
d_(MN) = sqrt(( 2h - h)^2 + ( 0 - k)^2)
d_(MN) = sqrt(h^2 + (- k)^2)
d_(MN) = sqrt(h^2 + k^2)