SOLUTION: Find the range for the measure of the third side of a triangle if the measures of the two sides are 10 feet & 16 feet.

Question 1084198: Find the range for the measure of the third side of a triangle if the measures of the two sides are 10 feet & 16 feet.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

We're given
first side = 10 ft
second side = 16 ft

Let x be the third side

The sum of any two sides must be larger than the other side
This is what the triangle inequality theorem is about.

(first side) + (second side) > third side
10 + 16 > x
26 > x
x < 26

(first side) + (third side) > second side
10 + x > 16
10 + x - 10 > 16 - 10
x > 6

(second side) + (third side) > first side
16 + x > 10
16 + x-16 > 10-16
x > -6

So we know
x > -6
x > 6
x < 26

The two inequalities x > -6 and x > 6 can be absorbed together to get x > 6. So we really have two inequalities which are
x > 6
x < 26

Combine them to form one compound inequality
x > 6 and x < 26
6 < x and x < 26
6 < x < 26

Therefore, the third side x is between 6 and 26.
The third side cannot be equal to 6.
The third side cannot be equal to 26.

As a shortcut you can use the following
a-b < c < a+b
where
a,b are the two known sides of the triangle (a > b)
c is the unknown side

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