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2. On a number line, the coordinates of [tex]\( P, Q, R \)[/tex] are 3, -5, and 6 respectively. State with reasons whether the following statements are true or false.

(i) [tex]\( d(P, Q) + d(Q, R) = d(P, R) \)[/tex]

(ii) [tex]\( d(P, R) + d(R, Q) = d(P, Q) \)[/tex]

(iii) [tex]\( d(R, P) + d(P, Q) = d(R, Q) \)[/tex]

(iv) [tex]\( d(P, Q) - d(P, R) = d(Q, R) \)[/tex]

Sagot :

GinnyAnswer
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
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