Question

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

Answer 1

0.5 in and 2 in.

Explanation:

We have:

`2a+b=15.7` subtract b from both sides

`2a+b-b=15.7-b`

`2a=15.7-b` divide both sides by 2

`(2a)/(2)=(15.7-b)/(2)`

`a=(15.7-b)/(2)\qquad\bold{(*)}`

We know:

`2a>b\qquad\bold{(**)}`

Subtitute (*) to (**):

`2\!\!\!\!\diagup\cdot(15.7-b)/(2\!\!\!\!\diagup)>b`

`15.7-b>b` add to both sides

`15.7-b+b>b+b`

`15.7>2b` divide both sides by 2

`(15.7)/(2)>(2)/(b)\\\\7.85>b\to \boxed{b<7.85}\ \text{and}\ \boxed{b>0}`

Only lengths 0.5in and 2in satisfy this inequality.

Answer 2

b=0.5 in

b=2 in

Explanation:

we know that

The perimeter of triangle is equal to

`2a+b=15`

Solve for a

`a=(15-b)/(2)` -----> equation A

Applying the Triangle Inequality Theorem

a+a > b

2a > b -----> inequality B

Verify each case

case 1) b=-2 in

This value not make sense, the length side cannot be a negative number

case 2) b=0 in

This value not make sense

case 3) b=0.5 in

substitute the value of b in the equation A and solve for a

`a=(15-0.5)/(2)=7.25\ in`

substitute the values of b and a in the inequality B

2a > b

2(7.25) > 0.5

14.50 > 0.5 -----> is true

therefore

b=0.5 in make sense for possible values of b

case 4) b=2 in

substitute the value of b in the equation A and solve for a

`a=(15-2)/(2)=6.5\ in`

substitute the values of b and a in the inequality B

2a > b

2(6.5) > 2

13 > 2 -----> is true

therefore

b=2 in make sense for possible values of b

case 5) b=7.9 in

substitute the value of b in the equation A and solve for a

`a=(15-7.9)/(2)=3.55\ in`

substitute the values of b and a in the inequality B

2a > b

2(3.55) >7.9

7.1 > 7.9 -----> is not true

therefore

b=7.9 in not make sense for possible values of b