Answer:
Case 1:
[tex]AB = 30[/tex]
[tex]BC = 50[/tex]
Case 2:
[tex]AB = 15.9[/tex]
[tex]BC = 36.7[/tex]
Case 3: Not possible
Step-by-step explanation:
Given
See attachment for illustration of each case
Required
Find AB and BC
Case 1:
Using Pythagoras theorem in ANB, we have:
[tex]AB^2 = AN^2 + BN^2[/tex]
This gives:
[tex]AB^2 = 24^2 + 18^2[/tex]
[tex]AB^2 = 576 + 324[/tex]
[tex]AB^2 = 900[/tex]
Take square roots of both sides
[tex]AB = \sqrt{900[/tex]
[tex]AB = 30[/tex]
To calculate BC, we consider ANC, where:
[tex]AC^2 = AN^2 + NC^2[/tex]
[tex]40^2 = 24^2 + NC^2[/tex]
[tex]1600 = 576 + NC^2[/tex]
Collect like terms
[tex]NC^2 = 1600 - 576[/tex]
[tex]NC^2 = 1024[/tex]
Take square roots
[tex]NC = \sqrt{1024[/tex]
[tex]NC = 32[/tex]
So:
[tex]BC = NC + BN[/tex]
[tex]BC = 32 + 18[/tex]
[tex]BC = 50[/tex]
Case 2:
Using Pythagoras theorem in ANB, we have:
[tex]AN^2 = AB^2 + BN^2[/tex]
This gives:
[tex]24^2 = AB^2 + 18^2[/tex]
[tex]576 = AB^2 + 324[/tex]
Collect like terms
[tex]AB^2 = 576 - 324[/tex]
[tex]AB^2 = 252[/tex]
Take square roots of both sides
[tex]AB = \sqrt{252[/tex]
[tex]AB = 15.9[/tex]
To calculate BC, we consider ABC, where:
[tex]AC^2 = AB^2 + BC^2[/tex]
[tex]40^2 = 252 + BC^2[/tex]
[tex]1600 = 252 + BC^2[/tex]
Collect like terms
[tex]BC^2 = 1600 - 252[/tex]
[tex]BC^2 = 1348[/tex]
Take square roots
[tex]BC = \sqrt{1348[/tex]
[tex]BC = 36.7[/tex]
Case 3:
This is not possible because in ANC
The hypotenuse AN (24) is less than AC (40)
