In a right triangle, the sum of the 2 legs are 23. The hypotenuse is 2 more than the longer leg. Find its dimensions.

Question

Needs to be done using the quadratic formula and I need let statements 

Answer 1

First define variables for the sides of the right triangle. 

 

Let the shorter leg = x

Let the longer leg = y

Let the hypotenuse = y + 2

 

Using these variables, we can write a system of equations.  The first equation will represent the sum of the two legs.  The other equation will represent the Pythagorean theorem.  Afterall, we are dealing with a right triangle.

 

x + y = 23                   eq1

 

x² + y² = (y + 2)²                 eq2

 

x² + y² = y² + 4y + 4

 

x² = 4y + 4                         eq2

 

 

Substitute eq1 into eq2 so that eq2 is in terms of x.

 

x² = 4(23 - x) + 4

 

x² = 92 - 4x + 4

 

 

Move all the terms to the left side of the equation to make the right side equal to zero.

 

x² + 4x - 96 = 0

 

 

This is your quadratic equation.  Use the quadratic formula to solve for x:

 

x = (-b ± √(b² - 4ac)) / 2a

 

where:

a = 1

b = 4

c = -96

 

Plug in these values into the formula.  You will get two x values.  Accept the positive x value.  Once you have your x value, plug it into the variables to find the dimensions.