Use the Triangle Inequality Theorem on the unknown side in each of the three triangles â–ł FGJ, â–ł GJH, and â–ł FJH
Between 4 and 24 units.
Practice makes perfect
To box in the perimeter of â–ł HGF we will use the Triangle Inequality Theorem on the unknown side in each of the three triangles â–ł FGJ, â–ł GJH, and â–ł FJH
Let's first examine the unknown side of â–ł FGJ. There are two cases for the length of FG.
It's the longest side.
It's not the longest side. Note that this makes FJ the longest side.
In the first case, the sum of FJ and GJ must be longer than FG. By the Triangle Inequality Theorem, we can write an inequality describing the length of FG.
5+3 > FG ⇒ FG< 8
In the second case, the sum of the lengths of GJ and FG must be greater than the length of FJ. Therefore, by the Triangle Inequality Theorem, we can write another inequality describing the length of FG.
FG+3>5 ⇒ FG>2
Now we have boxed in the length of FG.
If we do this for â–ł GJH, and â–ł FJH we get two more sets of inequalities.
&â–ł GJH
& 3+4>GH ⇒ GH< 7
& 3+GH > 4 ⇒ GH > 1
&â–ł FJH
& 4+5>FH ⇒ FH< 9
& 4+FH > 5 ⇒ FH > 1
Let's add these inequalities to our diagram.
By adding the lower boundaries of our three inequalities, we get the lower boundary of our perimeter, P. Similarly, if we add the upper boundaries of the inequalities, we get the upper boundary of our perimeter.
2+1+1 < P < 8+9+7 ⇒ 4 < P < 24
The perimeter is between 4 and 24 units.
