Certainly! Let's solve these two parts step-by-step.
### Part (i)
Given:
- Points are in a straight line: [tex]\( A - B - C \)[/tex]
- Distance between [tex]\( A \)[/tex] and [tex]\( C \)[/tex], [tex]\( d(A, C) = 17 \)[/tex]
- Distance between [tex]\( B \)[/tex] and [tex]\( C \)[/tex], [tex]\( d(B, C) = 6.5 \)[/tex]
We need to find the distance between [tex]\( A \)[/tex] and [tex]\( B \)[/tex], [tex]\( d(A, B) \)[/tex].
Since point [tex]\( B \)[/tex] is between [tex]\( A \)[/tex] and [tex]\( C \)[/tex] on the straight line, we can use the relationship:
[tex]\[
d(A, C) = d(A, B) + d(B, C)
\][/tex]
Given:
[tex]\[
17 = d(A, B) + 6.5
\][/tex]
To find [tex]\( d(A, B) \)[/tex], we subtract [tex]\( d(B, C) \)[/tex] from [tex]\( d(A, C) \)[/tex]:
[tex]\[
d(A, B) = 17 - 6.5
\][/tex]
[tex]\[
d(A, B) = 10.5
\][/tex]
So, the distance between [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is [tex]\( 10.5 \)[/tex].
### Part (ii)
Given:
- Points are in a straight line: [tex]\( P - Q - R \)[/tex]
- Distance between [tex]\( P \)[/tex] and [tex]\( Q \)[/tex], [tex]\( d(P, Q) = 3.4 \)[/tex]
- Distance between [tex]\( Q \)[/tex] and [tex]\( R \)[/tex], [tex]\( d(Q, R) = 5.7 \)[/tex]
We need to find the distance between [tex]\( P \)[/tex] and [tex]\( R \)[/tex], [tex]\( d(P, R) \)[/tex].
Since point [tex]\( Q \)[/tex] is between [tex]\( P \)[/tex] and [tex]\( R \)[/tex] on the straight line, we can use the relationship:
[tex]\[
d(P, R) = d(P, Q) + d(Q, R)
\][/tex]
Given:
[tex]\[
d(P, R) = 3.4 + 5.7
\][/tex]
[tex]\[
d(P, R) = 9.1
\][/tex]
So, the distance between [tex]\( P \)[/tex] and [tex]\( R \)[/tex] is [tex]\( 9.1 \)[/tex].
### Summary:
(i) [tex]\( d(A, B) = 10.5 \)[/tex]
(ii) [tex]\( d(P, R) = 9.1 \)[/tex]
