If a=5 and c=13, how do you find b?

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Answer 1 · 2015-10-17T21:23:47

Use Pythagoras and rearrange to find #b = 12#

Assuming we're dealing with a right angled triangle with legs of lengths #a#, #b# and hypotenuse of length #c#, Pythagoras theorem tells us:

#a^2+b^2 = c^2#

Subtracting #a^2# from both sides, we get:

#b^2 = c^2-a^2#

Then taking the square root of both sides, we get:

#b = sqrt(c^2 - a^2)#

We are told that #a = 5# and #c = 13#, so

#b = sqrt(13^2-5^2) = sqrt(169-25) = sqrt(144) = 12#

Bonus

The #5, 12, 13# triangle is the second one in a sequence of right angled triangles that starts with the #3, 4, 5# triangle.

#a = 2k + 3#

#b = (a^2 - 1) / 2 = 2k^2+6k+4#

#c = (a^2 + 1) / 2 = 2k^2+6k+5#

This gives us right angled triangles with sides:

#k=0:# #3, 4, 5#

#k=1:# #5, 12, 13#

#k=2:# #7, 24, 25#

#k=3:# #9, 40, 41#

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