a rectangle is defined by the vertices A(-6,5), B(12,-1), C(8,-13), and D(-10,-7). prove that the diagonals are equal in length
To prove that the diagonals of a rectangle are equal in length, we need to show that the lengths of both diagonals are equal. Let's calculate the lengths of the two diagonals using the distance formula.
The distance formula states that the distance between two points (x1, y1) and (x2, y2) is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's calculate the lengths of the diagonals AC and BD.
For diagonal AC:
Point A = (-6, 5)
Point C = (8, -13)
Using the distance formula, we get:
AC = sqrt((8 - (-6))^2 + ((-13) - 5)^2)
= sqrt(14^2 + (-18)^2)
= sqrt(196 + 324)
= sqrt(520)
= 2 * sqrt(130)
For diagonal BD:
Point B = (12, -1)
Point D = (-10, -7)
Using the distance formula, we get:
BD = sqrt((-10 - 12)^2 + (-7 - (-1))^2)
= sqrt((-22)^2 + (-6)^2)
= sqrt(484 + 36)
= sqrt(520)
= 2 * sqrt(130)
As we can see, both diagonals have the same length, AC = BD = 2 * sqrt(130). Therefore, we have proved that the diagonals of the given rectangle are equal in length.
Same idea as your next question.
Post your answers for verification if you wish.
http://www.jiskha.com/display.cgi?id=1288301772