Exam-style IB SL maths question

Question

The two pependicular sides of a right-angled triangle have lengths x+2 and 5x-3. The hypotenuse has length 4x+1. Find x.

(Please show working!)

Answer 1

Use the Pythagorean Theorem: a²+b²=c² where a and b are the lengths of the perpendicular sides and c is the hypotenuse.  In this case, a=x+2, b=5x-3, and c=4x+1

 

(x+2)² + (5x-3)² = (4x+1)²

 

x² + 4x + 4 + 25x² - 30x + 9 = 16x² + 8x + 1

 

10x² - 34x  + 12 = 0

 

Factors to:

 

(5x-2)(2x-6) = 0

 

x = 2/5 and 3

 

Check:

(x+2)² + (5x-3)² = (4x+1)²

 

For x = 2/5 = 0.4:

(2.4)² + (-1)2 = (2.6)²

5.76 + 1 = 6.76

6.76 = 6.76             CHECK!

 

For x = 3:

5² + 12² = 13²

25 + 144 = 169

169 = 169               CHECK!