SOLUTION: .What best describes the triangle whose corners are located at the points (1, 1), (2, 2), and (0, 16)? A. It is isosceles. B. It is a right triangle. C. It has area 22 sq

Question 635732: .What best describes the triangle whose corners are located at the points (1, 1), (2, 2), and (0, 16)?
A. It is isosceles.
B. It is a right triangle.
C. It has area 22 square units.
D. None of the above

Answer by jsmallt9(3759) (Show Source): You can put this solution on YOUR website!
To determine if the triangle is isoceles and/or right, we must know the lengths of the sides. And to find the area we will need at least one side. So we start by finding the lengths of the sides for the triangle.

TO find the lengths from coordinates we use the distance formula:

As we can see, the three sides all have different lengths. So the triangle is not isosceles. To find out if the triangle is a right triangle we see if the 3 sides fit the Pythagorean Theorem. The longest side is so it must be the hypotenuse (if it is a right triangle):

False! The triangle is not a right triangle.

I'm probably forgetting an easy way to find the area because the way I'm about to use is a bit of a pain. Area of a triangle is (1/2)*b*h. The base can be any side of the triangle. And the height is the length of the perpendicular from the opposite vertex to the line containing the base. I'm going to choose the side as our base. If we look at the coordinates of the vertices of that side, (1, 1) and (2, 2), I hope it is easy to understand that the line that contains these two is: y = x. The slope of the line y = x is 1.

The slope of any line perpendicular (like our height) will be the negative reciprocal of 1: -1. So our height is on a line with a slope of -1 and it passes through the opposite vertex: (0, 16). (0, 16) happens to be on the y-axis. So it is the y-intercept of the line that contains the height. So this line has a slope of -1 and a y-intercept of 16. The equation of this line is y = -x + 16.

Next we need to find the point where the line that contains the height intersect the line that contains our base. Use any method (Substitution, Linear Combination, others) to solve the system of y = x and y = -x + 16. You should get the point (8, 8).

Now we can find the length of the height:

If you have had any trouble following this, you might want to plot the vertices of the triangle on a graph, draw in the sides of the triangle, draw the lines that contain the base and height (y = x and y = -x+16) as dotted lines, and the posint where they intersect: (8, 8)

We're finally ready to find the area:

So the area is not 22.

The answer is D.
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