Let the length of each leg be x or some other variable. Then use the perimeter of 30 to find the measure of the triangle's base.
1/7
Practice makes perfect
Let x be the length of the triangle's legs. The sum of these two measures is therefore 2x. The perimeter of the triangle is 30 units. If we subtract 2x from 30, we will calculate the length of the triangle's base.
30-2x
By the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle is greater than the length of the third side. Let's use this theorem to form three inequalities true for the triangle.
x+x>30-2x
x+30-2x>x
30-2x+x>xNow we will solve these inequalities for x.
Inequality
Simplify
Solution Set
x+x>30-2x
4x>30
x>7.5
x+30-2x>x
30>2x
15>x
30-2x+x>x
30>2x
15>x
As we can see, the last two inequalities have the same solution sets. Thus, we can consider only one of these inequalities. Let's find the common solutions of the first two inequalities by graphing their solution sets.
The lines overlap on the segment from 7.5 to 15. Thus, the possible length of the legs is between 7.5 and 15.
7.5
Legs Length
Substitute
Base
8
30-2( 8)
14
9
30-2( 9)
12
10
30-2( 10)
10
11
30-2( 11)
8
12
30-2( 12)
6
13
30-2( 13)
4
14
30-2( 14)
2
We got 7 possible sets of the isosceles triangle's sides lengths. Out of them, only one makes the isosceles triangle equilateral with the length of each side of 10 units. Now we can calculate the probability of the triangle being equilateral using the Probability Formula.
Probability=Number of Favorable Outcomes/Number of Possible Outcomes
In our case, the number of possible outcomes is 7 and the number of favorable outcomes is 1. Let's substitute these numbers into the formula and calculate the probability.
Probability=1/7
The probability of the isosceles triangle with the perimeter of 30 units being equilateral is 17.
