Question

On a number line, co-ordinates of P, Q, R are 3,$-$ 5 and 6 respectively. State with reason whether the following statements are true or false.

(i) d(P,Q) + d(Q,R) = d(P,R)
(ii) d(P,R) + d(R,Q) = d(P,Q)

(iii) d(R,P) + d(P,Q) = d(R,Q)

(iv) d(P,Q) $-$ d(P,R) = d(Q,R)

Answer 1

The co-ordinates of points P and Q are 3 and −5 respectively. We know that 3 > −5.
Now, d(P, Q) = 3 − (−5) = 3 + 5 = 8
The co-ordinates of points Q and R are −5 and 6 respectively. We know that 6 > −5.
Now, d(Q, R) = 6 − (−5) = 6 + 5 = 11
The co-ordinates of points P and R are 3 and 6 respectively. We know that 6 > 3.
Now, d(P, R) = 6 − 3 = 3

(i) d(P, Q) + d(Q, R) = 8 + 11 = 19; d(P, R) = 3
So, d(P, Q) + d(Q, R) ≠ d(P, R)
Hence, the given statement is false.

(ii) d(P, R) + d(R, Q) = d(P, R) + d(Q, R) = 3 + 11 = 14; d(P, Q) = 8
So, d(P, R) + d(R, Q) ≠ d(P, Q)
Hence, the given statement is false.

(iii) d(R,P) + d(P,Q) = d(P, R) + d(P,Q) = 3 + 8 = 11; d(R,Q) = d(Q, R) = 11
So, d(R,P) + d(P,Q) = d(R,Q)
Hence, the given statement is true.

(iv) d(P,Q) $-$ d(P,R) = 8 − 3 = 5; d(Q,R) = 11
So, d(P,Q) $-$ d(P,R) ≠ d(Q,R)
Hence, the given statement is false.