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60
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For what value of a is there a right triangle with sides a+1, 6a, and 6a+1?

 

Thank you so much!!!

 Apr 19, 2020
 #1
avatar+44 
+3

Starting off, a right triangle can be expressed in form a^2 + b^2 = c^2. Its kind of obvious that 6a + 1 is the hypotenuse assuming a >= 1. your equation is (a + 1)^2 + 6a^2 = (6a + 1)^2. Expanding this out, it is a^2 + 2a + 1 + 36a^2 = 36a^2 + 12a + 1. Combining like terms, and subtracting 1 and 36a^2 from both sides, a^2 + 2a = 12a. this is fairly easy to solve, 12a - 2a = 10a, so a^2 = 10a. From here it is immediatly obvious that if a * a = a * 10, a must equal 10.

Checking,

11, 60, 61: 121 + 3600 = 3721, which is the square of 61, so this answer is correct.

NOTE: THIS ASSUMES a + 1 >= 1!

 Apr 19, 2020

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