△ABC  is a right triangle with right angle C. Side AC is 6 units longer than side BC . If the hypotenuse has length 52–√ units, find the length of AC.

Answer:AC = 7.12 unitsStep-by-step explanation:A right triangle has two legs and a hypotenuse. The hypotenuse is opposite the right angle. As Angle C is the right angle, then the triangle can be constructed as shown in the picture attached. The sides of the triangle have a relationship known as the Pythagorean Theorem a² + b² = c². In the theorem, the legs of the triangle are a and b while the hypotenuse is c. Substitute a = x, b = x+6, and c = √52. Simplify and solve.a² + b² = c²x² + (x+6)² = √52²x² + x² + 12x + 36 = 522x² + 12x - 16 = 0You can use the quadratic formula to solve by substituting a = 2, b = 12, and c = -16.The quadratic formula is [tex]x=\frac{-b+/-\sqrt{b^2-4ac} }{2a}[/tex]. Substitute and you'll have:[tex]x=\frac{-b+/-\sqrt{b^2-4ac} }{2a} =\frac{-12+/-\sqrt{12^2-4(2)(-16)} }{2(2)}=\frac{-12+/-\sqrt{144+128} }{4)}[/tex][tex]\frac{-12+/-\sqrt{272} }{4}=\frac{-12+/-16.5 }{4} = 1.12, 7.13[/tex]Only 1.12 is a solution since 7.13 will not satisfy the Pythagorean theoremSide AC is 6 units longer than side BC. This means x = BC and AC = x + 6. AC = 1.12 + 6 = 7.12