# Thread: Proving a triangle is right-angled.

1. ## Proving a triangle is right-angled.

Show that the triangle with vertices P(1,2,2), Q(7,-1,-1) and R(3,0,8) is right-angled.

My notes tell me that a triangle is only right angled if p.q = 0, but p.q = -11, p.r = 19, q.r = -29. I also tried to find out the individual angles and I got PQ = 120, PR = 42, QR= 118.

Am I making an obvious mistake or just doing something stupidly wrong? Thanks for your help :]

2. So

$P\left[\begin{array}{ccc}1\\2\\2\end{array}\right]$ $Q\left[\begin{array}{ccc}7\\-1\\-1\end{array}\right]$ $R\left[\begin{array}{ccc}3\\0\\8\end{array}\right]$

P.Q = (1.7) + (2.-1) + (2.-1) = 7 - 2 - 2 = 3
P.R = (1.3) + (2.0) + (2.8) = 3 + 16 = 19
Q.R = (7.3) + (-1.0) + (-1.8) = 21 - 8 = 12

I believe the most likely situation is that the question or your information is wrong

The question is exactly as it is in the book, strange. Thanks for proving that it isn't right-angled :]

4. Originally Posted by scofield131
Show that the triangle with vertices P(1,2,2), Q(7,-1,-1) and R(3,0,8) is right-angled.

My notes tell me that a triangle is only right angled if p.q = 0, but p.q = -11, p.r = 19, q.r = -29. I also tried to find out the individual angles and I got PQ = 120, PR = 42, QR= 118.

Am I making an obvious mistake or just doing something stupidly wrong? Thanks for your help :]
1. The legs of the triangle are $\overrightarrow{PQ}=(6,-3,-3)$ and $\overrightarrow{PR}=(2,-2,6)$

2. $\overrightarrow{PQ} \cdot \overrightarrow{PR} = 0$

3. Thus triangle PQR is a right triangle with the right angle at P.

5. Ahhh, I forgot about that. Thank you so much :]