Sure, let's go through each statement one by one and verify its validity.
The coordinates given are [tex]\( P = 3 \)[/tex], [tex]\( Q = -5 \)[/tex], and [tex]\( R = 6 \)[/tex].
First, let's calculate all the distances between these points:
- [tex]\( d(P, Q) \)[/tex] is the absolute difference between the coordinates of P and Q.
[tex]\[
d(P, Q) = |3 - (-5)| = |3 + 5| = 8
\][/tex]
- [tex]\( d(Q, R) \)[/tex] is the absolute difference between the coordinates of Q and R.
[tex]\[
d(Q, R) = |-5 - 6| = |-11| = 11
\][/tex]
- [tex]\( d(P, R) \)[/tex] is the absolute difference between the coordinates of P and R.
[tex]\[
d(P, R) = |3 - 6| = |-3| = 3
\][/tex]
- [tex]\( d(R, P) \)[/tex] is the absolute difference between the coordinates of R and P, which should be the same as [tex]\( d(P, R) \)[/tex] because distance is symmetric.
[tex]\[
d(R, P) = |6 - 3| = 3
\][/tex]
Now let's evaluate each statement:
(i) [tex]\( d(P, Q) + d(Q, R) = d(P, R) \)[/tex]
[tex]\[
8 + 11 = 3
\][/tex]
This is clearly false because 19 is not equal to 3.
(ii) [tex]\( d(P, R) + d(R, Q) = d(P, Q) \)[/tex]
[tex]\[
3 + 3 = 8
\][/tex]
This is false because 6 is not equal to 8.
(iii) [tex]\( d(R, P) + d(P, Q) = d(R, Q) \)[/tex]
[tex]\[
3 + 8 = 11
\][/tex]
This is true because 11 is equal to 11.
(iv) [tex]\( d(P, Q) - d(P, R) = d(Q, R) \)[/tex]
[tex]\[
8 - 3 = 11
\][/tex]
This is false because 5 is not equal to 11.
Thus, the statements evaluated are:
- (i) False
- (ii) False
- (iii) True
- (iv) False
