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Find the value of [tex]n[/tex] and [tex]Y Z[/tex] if [tex]Y[/tex] is between [tex]X[/tex] and [tex]Z[/tex].

[tex]X Y = 5n[/tex], [tex]Y Z = 2n[/tex], [tex]X Z = 91[/tex]

[tex]n = \quad[/tex]

[tex]Y Z = \quad[/tex]

Sagot :

GinnyAnsweridn
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]
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