Question

Two sides of a triangle have the following measures, Vind the range of possible measures for

the third side,
13) 8,7
14) 12,6
A) 1 B) 2 < x < 14
C) 1 D) 1
A) 8 B) 9 C) 8 < X < 18
D) 6 < X < 18

Answer 1

`(13)\ 1 <x < 15`

`(14)\ 6 <x < 18`

Explanation:

Question 13:

`a,b = 8,7` -- the two sides

Using triangle inequality theorem, we have:

`a + b > x`

`a + x > b`

`x + b > a`

So, we have:

`a + b > x`

`8 + 7 > x`

This gives:

`15 > x`

`a + x > b`

`8 + x > 7`

Collect and evaluate like terms

`x > -1`

`x + b > a`

`x + 7 > 8`

Collect and evaluate like terms

`x > 1`

Ignore the inequality with a negative value.

So, we have:

`x > 1` and
`15 > x`

Rewrite as:

`1< x` and
`x < 15`

Merge

`1 <x < 15`

Question 14:

`a,b = 12,6` -- the two sides

Using triangle inequality theorem, we have:

`a + b > x`

`a + x > b`

`x + b > a`

So, we have:

`a + b > x`

`12 + 6 > x`

This gives:

`18 > x`

`a + x > b`

`12 + x > 6`

Collect and evaluate like terms

`x > -6`

`x + b > a`

`x + 6 > 12`

Collect and evaluate like terms

`x > 6`

Ignore the inequality with a negative value.

So, we have:

`x > 6` and
`18 > x`

Rewrite as:

`6< x` and
`x < 18`

Merge

`6 <x < 18`