What are the possible lengths of the other two sides if one side if 17 inches long and the second side is 4 inches less than twice the third side?

Answer 1

For a triangle to exist the sum of the lengths of the shorter two sides must exceed the length of the other side.

Let the third side be x, then the second side is given by: 2x - 4

All sides must be greater than 0

→ 2x - 4 > 0

→ x > 2

The second and third sides are equal if:

2x - 4 = x

→ x = 4

If the third side is the longer of the two, then:

2x - 4 < x

→ x < 4

I

f the second side is the longer of the two, then:

2x - 4 > x

→ x > 4

Thus for 2 < x < 4 the third side is the longer of the two, and for x > 4, the second side is the longer.

There are two possibilities:

• The side of 17 inches is the longest side.

In this case the sum of the other two sides is greater than this:

x + 2x - 4 > 17

→ 3x > 21

→ x > 7

As x > 7 > 4, the longer of the two shorter sides is the second side and is at most 17 in so:

2x - 4 ≤ 17

→ x ≤ 10.5

Thus 7 < x ≤ 10.5

• One of the other sides is the longest side.

In this case, as 2x - 4 > x if x > 4, it is the second side of 2x - 4 that is the longest side, and:

2x - 4 ≥ 17

→ x ≥ 10.5

But this must be less than the sum of the other two sides, giving:

2x - 4 < 17 + x

→ x < 21

Thus 10.5 ≤ x < 21

Combining these two results gives:

7< x ≤ 10.5

10.5 ≤ x < 21

→ 7 < x < 21

Thus the other two sides are:

• second side is greater than 10 in and less than 38 in;
• third side is greater than 7 in and less than 21 in;

with the second equal to twice the third less 4 in.