Answer:
The length of the legs is 8.64cm and 14.64cm respectively
Step-by-step explanation:
I've added an attachment to aid my explanation.
At different intervals, I'll be making reference to it.
Given
[tex]AB = 17[/tex]
[tex]PAMC = 32[/tex]
[tex]PBMC = 25[/tex]
From the attachment, we have:
[tex]y + z = AB[/tex]
Since, M is the Midpoint
[tex]y = z = \½AB[/tex]
Substitute 17 for AB
[tex]y = z = \½ * 17[/tex]
[tex]y = z = 8.5[/tex]
Also, from the attachment
[tex]v + x + z = PAMC[/tex]
[tex]v + x + y = 32[/tex]
Substitute 8.5 for y
[tex]v + x + 8.5 = 32[/tex]
[tex]v + x = 32 - 8.5[/tex]
[tex]v + x = 23.5[/tex] --------- (1)
Also, from the attachment
[tex]v + w + z = 25[/tex]
Substitute 8.5 for z
[tex]v + w + 8.5 = 25[/tex]
[tex]v + w = 25 - 8.5[/tex]
[tex]v + w = 17.5[/tex] ----------- (2)
Subtract (2) from (1)
[tex]v - v + x - w = 23.5 - 17.5[/tex]
[tex]x - w = 6[/tex]
Make x the subject
[tex]x = 6 + w[/tex]
Apply Pythagoras Theorem:
We have that:
[tex]AB^2 = AC^2 + BC^2[/tex]
The above can be replaced with
[tex]17^2 = x^2 + w^2[/tex] (see attachment)
[tex]289 = x^2 + w^2[/tex]
Substitute 6 + w for x
[tex]289 = (6 + w)^2 + w^2[/tex]
[tex]289 = 36 + 12w + w^2 + w^2[/tex]
[tex]289 - 36 = 12w + 2w^2[/tex]
[tex]253 = 12w + 2w^2[/tex]
Reorder
[tex]2w^2 + 12w - 253 = 0[/tex]
Solve using quadratic equation:
[tex]w = \frac{-b \± \sqrt{b^2 - 4ac}}{2a}[/tex]
Where
[tex]a = 2[/tex]
[tex]b = 12[/tex]
[tex]c = -253[/tex]
[tex]w = \frac{-12 \± \sqrt{12^2 - 4 * 2 * -253}}{2 * 2}[/tex]
[tex]w = \frac{-12 \± \sqrt{144 + 2024}}{4}[/tex]
[tex]w = \frac{-12 \± \sqrt{2168}}{4}[/tex]
[tex]w = \frac{-12 \± 46.56}{4}[/tex]
Split:
[tex]w = \frac{-12 + 46.56}{4}[/tex] or [tex]w = \frac{-12 - 46.56}{4}[/tex]
[tex]w = \frac{34.56}{4}[/tex] or [tex]w = \frac{-58.56}{4}[/tex]
[tex]w = 8.64[/tex] or [tex]w = -14.64[/tex]
But length can't be negative
So:
[tex]w = 8.64[/tex]
Recall that: [tex]x = 6 + w[/tex]
[tex]x = 6 + 8.64[/tex]
[tex]x = 14.64[/tex]
Hence, the length of the legs is 8.64cm and 14.64cm respectively
