An isosceles triangle has two sides of equal length, a, and a base, b. The perimeter of the triangle is 15.7 inches, so the equation to solve is 2a + b = 15.7. If we recall that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side, which lengths make sense for possible values of b? Select two options. –2 in. 0 in. 0.5 in. 2 in. 7.9 in.
The Correct Answer and Explanation is:
To solve the problem, we start with the given perimeter equation of the isosceles triangle:2a+b=15.72a + b = 15.72a+b=15.7
This equation tells us that the sum of the two equal sides, 2a2a2a, plus the base bbb, is 15.7 inches. Rearranging the equation, we can express aaa in terms of bbb:2a=15.7−b⇒a=15.7−b22a = 15.7 – b \quad \Rightarrow \quad a = \frac{15.7 – b}{2}2a=15.7−b⇒a=215.7−b
Now, consider the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In an isosceles triangle with side lengths aaa, aaa, and bbb, the triangle inequality gives us three conditions:
- a+a>ba + a > ba+a>b or 2a>b2a > b2a>b
- a+b>aa + b > aa+b>a which simplifies to b>0b > 0b>0
- a+b>aa + b > aa+b>a again simplifies to b>0b > 0b>0
From these, we mainly need to ensure that:
- b>0b > 0b>0
- 2a>b2a > b2a>b
Let’s test each option to see which values of bbb satisfy the conditions.
Option: -2 in
This is not valid. Side lengths of a triangle must be positive. Also, b>0b > 0b>0 fails.
Option: 0 in
Again, not valid. The base cannot be zero. b>0b > 0b>0 fails.
Option: 0.5 in
Try plugging into the equation:
a=15.7−0.52=15.22=7.6a = \frac{15.7 – 0.5}{2} = \frac{15.2}{2} = 7.6a=215.7−0.5=215.2=7.6
Then check:
2a=15.2>0.5=b2a = 15.2 > 0.5 = b2a=15.2>0.5=b
Both triangle inequalities are satisfied. Valid
Option: 2 in
a=15.7−22=13.72=6.85a = \frac{15.7 – 2}{2} = \frac{13.7}{2} = 6.85a=215.7−2=213.7=6.85
Then 2a=13.7>2=b2a = 13.7 > 2 = b2a=13.7>2=b
Also satisfies triangle inequalities. Valid
Option: 7.9 in
a=15.7−7.92=7.82=3.9a = \frac{15.7 – 7.9}{2} = \frac{7.8}{2} = 3.9a=215.7−7.9=27.8=3.9
Then 2a=7.8<7.9=b2a = 7.8 < 7.9 = b2a=7.8<7.9=b
Fails the condition 2a>b2a > b2a>b. Invalid
Correct answers: 0.5 in and 2 in
These values satisfy all the triangle inequality conditions and result in positive side lengths.
