SOLUTION: A triangle has the coordinates (2, 7), (3, 5) and (1,0) Is this a right triangle?

Algebra ->  Triangles -> SOLUTION: A triangle has the coordinates (2, 7), (3, 5) and (1,0) Is this a right triangle?      Log On


   

Question 1203533: A triangle has the coordinates (2, 7), (3, 5) and (1,0)
Is this a right triangle?
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3835) About Me  (Show Source):
You can put this solution on YOUR website!

Answer: No, this is not a right triangle.

Explanation

Label the points A,B,C. The order of which doesn't matter.
A = (2,7)
B = (3,5)
C = (1,0)

I'll use the distance formula to compute how far it is from A to B.
This will give us the length of segment AB.

A = (x1,y1) = (2,7) and B = (x2,y2) = (3,5)
d = distance from A to B = length of segment AB
d+=+sqrt%28+%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2+%29

d+=+sqrt%28+%282-3%29%5E2+%2B+%287-5%29%5E2+%29

d+=+sqrt%28+%28-1%29%5E2+%2B+%282%29%5E2+%29

d+=+sqrt%28+1+%2B+4+%29

d+=+sqrt%28+5+%29

d+=+2.236068

Segment AB is exactly sqrt%285%29 units long, which approximates to about 2.236068 units.

Use that same formula to find these other segment lengths:
BC+=+sqrt%2829%29+=+5.385165
AC+=+sqrt%2850%29+=+5%2Asqrt%282%29+=+7.071068
The decimal values are approximate.

Once we know the side lengths, we can plug them into the pythagorean theorem equation.
If we get the same thing on both sides, then we conclude the triangle is a right triangle.



%28AB%29%5E2%2B%28BC%29%5E2=%28AC%29%5E2 The hypotenuse is always the longest side. In this case, the longest side is AC = sqrt(50) = 7.071068 roughly.

%28sqrt%285%29%29%5E2%2B%28sqrt%2829%29%29%5E2=%28sqrt%2850%29%29%5E2

5%2B29=50

34=50

Because we ended up with a false equation, it means that %28AB%29%5E2%2B%28BC%29%5E2=%28AC%29%5E2 is also false.
Furthermore, we do NOT have a right triangle.

-------------------------

Here's another approach that might be more simple depending on your viewpoint.

Once again we have these three given points
A = (2,7)
B = (3,5)
C = (1,0)

Let's compute the slope of line AB.

A = (x1,y1) = (2,7) and B = (x2,y2) = (3,5)


m+=+%28y%5B2%5D+-+y%5B1%5D%29%2F%28x%5B2%5D+-+x%5B1%5D%29

m+=+%285+-+7%29%2F%283+-+2%29

m+=+%28-2%29%2F%281%29

m+=+-2
The slope of line AB is -2.

If you use the slope formula for the other sides, then:
slope of BC = 5/2 = 2.5
slope of AC = 7

Then multiply each possible slope pairing.
(slope AB)*(slope BC) = (-2)*(2.5) = -5
(slope AB)*(slope AC) = (-2)*(7) = -14
(slope BC)*(slope AC) = (2.5)*(7) = 17.5

None of the pairs multiply to -1.
Therefore, none of the pairs of lines are perpendicular and we do NOT have a right triangle.

There might be other approaches. Feel free to explore other alternatives.

Answer by ikleyn(53765) About Me  (Show Source):
You can put this solution on YOUR website!
.
A triangle has the coordinates (2, 7), (3, 5) and (1,0).
Is this a right triangle?
~~~~~~~~~~~~~~~~~ 

Let the points be A= (2,7),  B= (3,5), C= (1,0).


Then the three vectors are

    AB = (3-2,5-7) = (1,-2)

    AC = (1-2,0-7) = (-1,-7)

    BC = (1-3,0-5) = (-2,-5)


The trianle is right-angled if and only if one of the three scalar product of these three vectors is zero.
So, calculate the scalar products

    AB * AC = 1*(-1) + (-2)*(-7) = -1 + 14 = 13;

    AB * BC = 1*(-2) + (-2)*(-5) = -2 + 10 = -8;

    AC * BC = (-1)*(-2) + (-7)*(-5) = 2 + 35 = 37.


As you see, no one scalar product is zero - so we conclude that the triangle is not a right-angled triangle.

Solved.