Expert: Paul Klarreich - 10/12/2010

Question
Hi, can you please help me with this?

A right triangle has an area of 96 ft2  and a hypotenuse 20 ft long. What are the lengths of its other two sides?

And also,

The sum of two numbers is twice their difference. The larger number is 4 more than twice the smaller. Find the numbers.

Thanks!

Answer
Questioner:   Kristin
Country:  United States
Category:  Advanced Math
Private:  No
 
Subject:  Precalculus
Question:  Hi, can you please help me with this?

A right triangle has an area of 96 ft2  and a hypotenuse 20 ft long. What are the lengths of its other two sides?

And also,

The sum of two numbers is twice their difference. The larger number is 4 more than twice the smaller. Find the numbers.

Thanks!
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Start by:

Let one leg of the triangle be  x
 
Let the other leg be  y.

Now use "has an area of 96 ft2" and the formula, which you will look up, for the area of a triangle, to write an equation:

xy = 192  <you decide why>
or y = 192/x

Then use the pythagorean theorem to write another equation.

x^2 + y^2 = 20^2

Substitute:

x^2 + 192^2/x^2 = 20^2

x^4 + 192^2 = 400 x^2

x^4 -  400x^2 + 192^2 = 0

Some routine arithmetic gives you

x = 16, x = 12,
which means y = 12 or y = 16  (they are symmetric)

.................................
For the other, write:

Let the larger number be  x

Let the smaller number be  y

From the sentence "The sum of two numbers is twice their difference.", write an expression for "The sum of two numbers" using x and y.

then write something using x and y for "is twice their difference."

Were you careful?  Did you parenthesize correctly?

Now you have an equation in x and y.  You need another.

Now from the sentence "The larger number is 4 more than twice the smaller"  you can write another:

Write something for "The larger number"
Write something for "is"
Write something for "4 more than"
Write something for "twice the smaller"

Put it all together now.