Question

An isosceles triangle has a base 10.6 units long. If the congruent side lengths have measure of the first decimal place, what is the shortest possible length of the side?

Answer 1

Answer: 5.4

Explanation below

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We have a triangle with sides a, b, c such that

• a = x
• b = x
• c = 10.6

By the triangle inequality theorem, we can say:

b-a < c < b+a

x-x < 10.6 < x+x

0 < 10.6 < 2x

10.6 < 2x

(10.6)/2 < x

5.3 < x

x > 5.3

Notice that none of these inequality signs have "or equal to" as part of them. So writing x > 5.3 means that we could have the following x values

• x = 5.6
• x = 5.5
• x = 5.4
• x = 5.35
• x = 5.34
• x = 5.31
• x = 5.3001
• x = 5.300001

and so on. Simply pick anything that's larger than 5.3 and that will work for x. We can't pick 5.3 itself. As that list goes on, we steadily get closer to 5.3 but not actually arrive to it. Therefore, there is no smallest value x can take on. No matter what value you pick, there's always something smaller and still makes x > 5.3 true.

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However, your teacher mentioned "measure of the first decimal place" which means we'll round to the nearest tenth.

We can't round to 5.3 since x > 5.3 turns into 5.3 > 5.3 which is false.

The next best thing is to round to 5.4, and we can see 5.4 > 5.3 is true.

Therefore, x = 5.4 is the shortest possible length where we are rounding to the nearest tenth (one decimal place).

Answer 2

5.4 units long.

Explanation:

Each congruent side must be greater than half the base length (5.3 units)