Find the length of each side

Question

A scalene triangle has 3 sides of different lengths. One of the sides is one less twice the shortest side. The longest side is five times the short side increased by six. The perimeter of the triangle is 101 cm. Find the length of each side.

Answer 1

Generally with these kind of questions, the goal is to express everything as equations, then by simplifying the equations, we can get the values of the individual variables.

To start, we have a scalene triangle with 3 different side lengths. Let's write the side lengths as a,b,c.

The first piece of information is that one side is one less than twice the shortest side. Let's write that statement as an equation:

b (one side)

= (is)

-1 (one less)

2×a (twice the shortest side)

So b = 2a - 1

The next piece of information is the longest side is five times the short side increased by six.

This can be expressed as c = 5a + 6

The last piece of information is that the perimeter of the triangle is 101 cm. The perimeter if a triangle is the sum of all sides.

So a + b + c = 101

So now we have all three statements as equations:

b = 2a - 1

c = 5a + 6

a + b + c = 101

The objective when we have a system of equations like this is to find a way to isolate one variable. From there, we can solve for that variable easily.

Don't worry about knowing how or which variable to isolate, with experience you will know which variable works best. And in the worst case, you can always try each one until you find something that works.

We see that the first equation expresses b using a, and the second equation expresses c using a. That means we can replace the b and c in the third equation using a.

a + b + c = 101

a + (2a - 1) + c = 101, since b = 2a - 1, we can replace b using that equation here.

a + (2a - 1) + (5a + 6) = 101, since c = 5a + 6, we can replace c using that equation here.

Now we simplify the equation by combining like terms:

a + 5a + 6 + 2a - 1 = 101

a + 5a + 2a + 6 - 1 = 101

8a + 5 = 101

Now we want to isolate a

8a + 5 = 101

8a = 96

a = 12

Wonderful, now that we have a, we can plus a back into the original equations to get sides b and c.

b = 2a - 1

b = 2(12) - 1

b = 24 - 1

b = 23

c = 5a + 6

c = 5 (12) + 6

c = 60 + 6

c = 66

So now we have all three sides:

a = 12, b = 23, c = 66

As a final check, we can make sure the third equation is also true (the perimeter of the triangle is 101).

a + b + c = 101

(12) + (23) + (66) = 101

101 = 101

Great we did it!