Question

Assertion ($A$): $P(3,5,4)$ , $Q(5,7,5)$ , $R(6,5,7)$ are vertices of a triangle whose orthocentre is $(5,7,5)$.
Reason ( R): In a right-angled triangle the right-angled vertex is orthocentre:

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true but R is not he correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Answer 1

The correct option is A Both A and R are true and R is the correct explanation of A

Given, $P(3,5,4),Q(5,7,5),R(6,5,7)$

The dr's of $PQ$ are $(2,2,1)$
The dr's of $QR$ are $(1,-2,2)$
$PQ.QR=2-4+2=0$
Hence, $PQR$ is a right angle triangle at $Q$.

So it's orthocentre will lie at the point $Q$ itself.

Therefore, both Assertion and Reason are correct. And Reason is correct explanation of Assertion.