Answer:
[tex]NP = 24.18[/tex]
Step-by-step explanation:
Given
[tex]NK = 7x - 1[/tex]
[tex]NM = 10x - 13[/tex]
[tex]KM = 24[/tex]
See attachment
Required
Length of NP
First, calculate the value of x
[tex]NK = NM[/tex]
So:
[tex]7x -1 = 10x - 13[/tex]
Collect like terms
[tex]10x - 7x = 13 - 1[/tex]
[tex]3x =12[/tex]
Solve for x
[tex]x = 4[/tex]
To solve for NP, we consider right triangle [tex]\triangle NKP[/tex]
Where
[tex]NK = 7x - 1[/tex]
[tex]NK = 7 * 4 - 1[/tex]
[tex]NK = 27[/tex]
[tex]KP =\frac{1}{2}KM[/tex]
[tex]KP =\frac{1}{2} * 24[/tex]
[tex]KP =12\\[/tex]
Using Pythagoras theorem, we have:
[tex]NK^2 = KP^2 + NP^2[/tex]
[tex]27^2 = 12^2 + NP^2[/tex]
[tex]729 = 144 + NP^2[/tex]
Collect like terms
[tex]NP^2 = 729 - 144[/tex]
[tex]NP^2 = 585[/tex]
Take square roots
[tex]NP = \sqrt{585[/tex]
[tex]NP = 24.18[/tex]
