word problem

Question

The length of the longer leg of a right triangle is 4 inches
more than twice the length of the shorter leg. The length of the hypotenuse is 6 inches
more than twice the length of the shorter leg. Find the side lengths of the triangle.

Answer 1

Hello, Brittany

 

Let's start by translating the given information into the universal second language which is math.

 

Four more than twice a number is 2x+ 4.... Twice means two times. And not 2^2... Be careful... 

 

Six more than twice a number is 2x+6.  Or you can write this as 6+ 2x.  They are the same, just like 2+3 is the same as 3+2. As you might recall this is the Commutative Property of Addition.  

 

Now, that we translated into algebra, we are ready to set the problem up;

Let x represent the length of the shorter leg.

Let 2x+4 represent the longer leg of the right triangle.

Let 2x+ 6 represent the hypotenuse

 

Next, we are going to use Pythagoras Theorem, a^2+b^2=c^2, where a and b are the legs of the right triangle, and c is the hypotenuse.  It does not matter whether a or b represent the smaller leg.  So, what this is telling us is to square both legs... add their sum.  And make that equal to the hypotenuse squared.  Ok, Let's start by squaring each leg.

 

Shortest leg is x, so when you are square, we get x^2.. Ok, easy enough.

 

Longest leg is 

(2x+4)^2              Squaring the longest leg

(2x+4)(2x+4)= 4x^2+8x+8x+16

4x^2+16x+16     This is the longest leg.

 

The hypotenuse is 

(2x+6)^2.              Squaring it

(2x+6)(2x+6)= 4x^2+12x+12x+36= 4x^2+ 24x+ 36

 

Now, we are almost ready to solve it. Remember, we need to add the squared legs.  So, let's do that.

 

x^2+ [4x^2+16x+ 16]= 5x^2+16x+16

This is the sum of the legs squared. 

 

Ok, Brittney, we are almost done.  Make the previous answer equal to the hypotenuse square, and solve for x. This is what we have

 

5x^2+16x+16= 4x^2+ 24x+36       Subtract 4x^2 from both sides

-4x^2               -4x^2

x^2+16x+16= 24x+36                    Subtract 24x from both sides

         -24x        -24x

x^2-8x+ 16= 36                             Subtract 36 from both sides

             -36   -36

x^2-8x-20=0                                  Factoring 

(x-10)(x+2)=0

x= 10 or x=-2

 

We cannot use x=-2 because length cannot be negative.  So, x= 10

The shortest side is x, so it is 10.

The longest side is 2x+ 4, so it is 24

The hypotenuse is 2x+6, so it is 26

 

This is a multiple of 5,12,13...which is a Pythagorean triplet. 

 

Well, it is kind of long, but I am glad to help out.  Please, don't forget to rate my answer.

 

D.Y. Taylor