system of equations.

Question

A rectangular lot has an area of 60m². The length of a diagonal of the lot is 13m. What are the length and width of the lot. us equadratic formula

Answer 1

When a diagonal is drawn across a rectangle, two congruent right triangles are formed.  The legs of the triangle are equivalent to the sides of the rectangle.  Since we have a right triangle, we can use the Pythagorean theorem.

 

Let leg one = x

Let leg two = y

 

Area of rectangle is   length × width

 

 

Knowing this we can set up equations to find the two legs. 

 

xy = 60                eq1

 

x² + y² = 13²

x² + y² = 169          eq2

 

 

Substitute eq2 into eq1.  We get everything in terms of x.

 

x √(169 - x²) = 60

 

 

Divide both sides of equation by x.

 

√(169 - x²) = 60/x

 

 

Square both sides of equation.

 

169 - x² = 3600 / x²

 

 

Multiply both sides of equation by x².

 

x²(169 - x²) = 3600

 

- x⁴  + 169x² - 3600 = 0

 

This will be difficult to solve by hand.  Therefore, we will use the equation solver on the TI calculator, if you are allowed to.

Press MATH, enter 0, and put in the equation obtained here.  After that, hit enter.  You will see a new screen that will allow to guess a solution. Type in any number next to x=  (positive and negative will grant you two different solutions).  Press ALPHA and hit ENTER.

 

You will have two solutions.  x = 5  and x = -5

 

Accept the positive value of x, which is 5.

 

Next, substitute this value of x into eq2 to find y.

 

y = √(169 - x²)

y = √(169 - 5²)

y = 12

 

Our two legs are 5m and 12m.  So the length and width are 5m and 12m.