Question

a triangle has vertices d(6,1), e(2,3) and f(-1,-3). show that triangle DEF is a right angle triangle and identify the right angle triangle

Answer 1

We'll need the slope formula which is

`m = (y_(2) - y_(1))/(x_(2) - x_(1))\\\\`

We subtract the y values together, and divide that over the difference in the x values when subtracted in the same order.

Let's find the slope of line DE

`D = (x_1,y_1) = (6,1) \text{ and } E = (x_2,y_2) = (2,3)\\\\m = (y_(2) - y_(1))/(x_(2) - x_(1))\\\\m = (3 - 1)/(2 - 6)\\\\m = (2)/(-4)\\\\m = -(1)/(2)\\\\`

The slope of line DE is -1/2.

Next, compute the slope of line EF

`E = (x_1,y_1) = (2,3) \text{ and } F = (x_2,y_2) = (-1,-3)\\\\m = (y_(2) - y_(1))/(x_(2) - x_(1))\\\\m = (-3 - 3)/(-1 - 2)\\\\m = (-6)/(-3)\\\\m = 2\\\\`

The slope of line EF is 2.

Lastly, compute the slope of line FD

`F = (x_1,y_1) = (-1,-3) \text{ and } D = (x_2,y_2) = (6,1)\\\\m = (y_(2) - y_(1))/(x_(2) - x_(1))\\\\m = (1 - (-3))/(6 - (-1))\\\\m = (1 + 3)/(6 + 1)\\\\m = (4)/(7)\\\\`

The slope of line FD is 4/7.

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To recap everything so far, we found the following:

• slope of DE = -1/2
• slope of EF = 2
• slope of FD = 4/7

The product of the first two slopes gets us (-1/2)*(2) = -1 showing that DE is perpendicular to EF.

Perpendicular slopes multiply to -1 as long as neither line is vertical nor horizontal.

Since DE is perpendicular to EF, this proves we have a 90 degree angle at point E.

Therefore triangle DEF is a right triangle.