Word Problem Triangle Question

Question

The perimeter of a triangle is 51 centimeters. The longest side is 1 centimeter less than the sum of the other sides. Twice the shortest side is 9 centimeters less than the longest side. Find the lengths of each side of the triangle.

Answer 1

Assume l the longest side; lo and ls (the shortest side) the others sides of the given rectangle.

 

lo + ls - 1 + lo + ls = 51

 

2 ls + 9 = lo + ls - 1

 

Two equations, two unknowns lo and ls ⇒ SOLVABLE!

 

2 lo + 2 ls = 52

   lo  -    ls = 10 

2 lo  - 2 ls = 20

 

4 lo           =  72      ===>    

 

lo = 18 centimeters   ===> ls = 8 centimeters  and l = 18 + 8 - 1 = 25 centimeters. 18 + 8 + 25 = 51

Answer 2

The perimeter of a triangle equals the sum of the lengths of the three sides.

 

Let's let the shortest length = X

Let's let the middle length = Y

Let's let the longest length = Z

 

Three variables require three equations to solve.

 

Now it's time to translate the English question into math:

 

1) The perimeter is 51 centimeters :  X + Y + Z = 51

2) The longest side is 1 centimeter less than the sum of the other sides:  Z = X + Y - 1

3) Twice the shortest side is 9 centimeters less than the longest side:  2X = Z - 9

 

Rewriting the first equation:  X + Y = 51 - Z

Now substitute 51 - Z for X + Y in the second equation:

 

Z = X + Y - 1

Z = 51 - Z - 1

2Z = 50

Z = 25

YAY! We got one variable's value!

 

Now plug this value of Z into the third equation to find the value of X:

2X = Z - 9

2X = 25 - 9

2X = 16

X = 8

 

YAY again!  We have the values of two variables!

 

Now plug in the values of X and Z into the first equation to find the value of Y:

 

X + Y + Z = 51

8 + Y + 25 = 51

Y + 33 = 51

Y = 18

 

Another way to solve the three equations is to rewrite equation 2:  X + Y - Z = 1

  Add it to equation 1                                                                   X + Y + Z = 51    

                                                                                                 ---------------------

                                                                                                  2X + 2Y = 52

              Divide both sides by 2-->                                                X + Y = 26

 

             Plug this into equation 1-->                 X + Y + Z = 51

                                                                      26 + Z = 51

                                                                         Z = 25

 

Now find X and Y like we did before.

 

Confirm these answers by plugging into the original question.

 

Dan 

 

Answer 3

a=longest side

b=middle side

c=shortest side

a=b+c-1 which implies b=a-c+1

2c=a-9 which implies c=(a-9)/2

P=a+b+c

51=a+(a-c+1)+c

51=a+a-c+1+c

51=2a+1

50=2a

a=25

c=(25-9)/2

c=16/2

c=8

b=a-c+1

b=25-8+1

b=18

a=25 cm, b=18 cm, c=8 cm