SOLUTION: The vertices of the triangle are A(6,1), B(6,7) and C(10,7) show that the triangle is right angled and find it's sides?

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Question 446465: The vertices of the triangle are A(6,1), B(6,7) and C(10,7) show that the triangle is right angled and find it's sides?
Found 2 solutions by mananth, ikleyn:
Answer by mananth(16949) About Me  (Show Source):
You can put this solution on YOUR website!
x1 y1 x2 y2
d= sqrt%28%28y2-y1%29%5E2%2B%28x2-x1%29%5E2%29
6 1 6 7
d(AB)= sqrt%28%287-1%29%5E2%2B%286-6%29%5E2%29
d= sqrt%28%286%29%5E2%09%2B%280%29%5E2%29
d= sqrt%28%2836%29%29
d(AB)= 6.00
x1 y1 x2 y2
d(BC)= sqrt%28%28y2-y1%29%5E2%2B%28x2-x1%29%5E2%29
10 7 6 7
d= sqrt%28%287-7%29%5E2%2B%286-10%29%5E2%29
d=sqrt%28%280%29%5E2%2B%28-4%29%5E2%29
d= sqrt%28%2816%29%29
d(BC)= 4.00
----
x1 y1 x2 y2
d(CA)= sqrt%28%28y2-y1%29%5E2%2B%28x2-x1%29%5E2%29
10 7 6 1
d= sqrt%28%281-7%29%5E2%2B%286-10%29%5E2%29
d= sqrt%28%28-6%29%5E2%2B%28-4%29%5E2%29
d= sqrt%28%2852%29%29
d(CA)= 7.215

d(AB)^2+d(BC)^2= d(AC)^2 if it is a right triangle
6^2+4^2= 7.215^2
52=52

Answer by ikleyn(53426) About Me  (Show Source):
You can put this solution on YOUR website!
.
The vertices of the triangle are A(6,1), B(6,7) and C(10,7) show that the triangle is right angled and find it's sides?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~


            It is so simple . . . - much simpler than you think. 


The side AB is vertical line x = 6,  since points A and B have the same x-coordinate 6.

The side BC is horizontal line y = 7, since points B and C have the same y-coordinate 7.



        Therefore, triangle ABC is a right-angled triangle.



The length of side AB is the difference y-coordinates points A and B  |7-1| = 6 units.

The length of side BC is the difference x-coordinates points B and C  |10-6| = 4 units.

The hypotenuse  AC has the length  sqrt%286%5E2%2B4%5E2%29 = sqrt%2836%2B16%29 = sqrt%2852%29 units.

At this point, the problem is solved completely: all questions are answered.

You do not need to make complicated reasoning or complicated calculations, as @mananth does.


This problem teaches you to retrieve out geometric information from coordinates
of given points in coordinate plane.