Sure! Let's break down the problem step by step.
We are given three important pieces of information about points [tex]\(X\)[/tex], [tex]\(Y\)[/tex], and [tex]\(Z\)[/tex]:
1. The distance between [tex]\(X\)[/tex] and [tex]\(Y\)[/tex] is [tex]\(XY = 5n\)[/tex].
2. The distance between [tex]\(Y\)[/tex] and [tex]\(Z\)[/tex] is [tex]\(YZ = 2n\)[/tex].
3. The total distance between [tex]\(X\)[/tex] and [tex]\(Z\)[/tex] is [tex]\(XZ = 91\)[/tex].
We are asked to find the value of [tex]\(n\)[/tex] and the distance [tex]\(YZ\)[/tex].
Given that [tex]\(Y\)[/tex] is between [tex]\(X\)[/tex] and [tex]\(Z\)[/tex], we can write the following equation based on the distances:
[tex]\[ XY + YZ = XZ \][/tex]
Substituting the given distances:
[tex]\[ 5n + 2n = 91 \][/tex]
Combine like terms on the left side:
[tex]\[ 7n = 91 \][/tex]
To solve for [tex]\(n\)[/tex], divide both sides of the equation by 7:
[tex]\[ n = \frac{91}{7} \][/tex]
[tex]\[ n = 13 \][/tex]
Now, we need to find the value of [tex]\(YZ\)[/tex].
Since [tex]\(YZ = 2n\)[/tex], substitute the value of [tex]\(n\)[/tex] we just found:
[tex]\[ YZ = 2n \][/tex]
[tex]\[ YZ = 2 \times 13 \][/tex]
[tex]\[ YZ = 26 \][/tex]
Thus, the value of [tex]\(n\)[/tex] is [tex]\(13\)[/tex] and the distance [tex]\(YZ\)[/tex] is [tex]\(26\)[/tex].
In conclusion:
[tex]\[ n = 13 \][/tex]
[tex]\[ YZ = 26 \][/tex]
