SOLUTION: Either Prove of Disprove the given statement: The points (3,3),(3,7) and (9,3) are the vertices of a right triangle.

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Question 204739: Either Prove of Disprove the given statement:
The points (3,3),(3,7) and (9,3) are the vertices of a right triangle.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
There are two ways to do this.
1) Show that the segments fit the Pythagorean theorem equation: a%5E2+%2B+b%5E2+=+c%5E2. In order to do this we need to find the lengths of these segments. The lengths of the segments can be found using the distance formula: d+=+sqrt%28%28x%5B2%5D+-+x%5B1%5D%29%5E2+%2B+%28y%5B2%5D+-+y%5B1%5D%29%5E2%29.



We can fit these squared lengths into the Pythagorean equation:
%28sqrt%2816%29%29%5E2+%2B+%28sqrt%2836%29%29%5E2+=+%28sqrt%2852%29%29%5E2
+16+%2B+36+=+52+ (Check!)
so these points do make a right triangle.

A second way, which is often easier, is to find the slopes of the segments using the slope formula: m+=+%28y%5B2%5D+-+y%5B1%5D%29%2F%28x%5B2%5D+-+x%5B1%5D%29
m%5B1%5D+=+%283-7%29%2F%283-3%29+=+-4%2F0 which is undefined
m%5B2%5D+=+%283-3%29%2F%283-9%29+=+0%2F%28-6%29+=+0
m%5B3%5D+=+%283-9%29%2F%287-3%29+=+%28-6%29%2F4+=+-3%2F2
When lines are perpendicular (IOW form a right angle), their slopes are usually negative reciprocals of each other. At first glance we do not see negative reciprocal slopes so we might think there are no perpendiculars. But additional thought should reveal that we do have perpendiculars. A slope of zero means a horizontal line. An undefined slope means a vertical line. Horizontal and vertical are perpendicular.

So either way we find that the three points do form a right triangle.