Find the length of the shortest straw

Question

Marlene has three straws two straws have the length shown. She does not know the length of the shortest straw but when she forms a triangle with all three the triangle is obtuse which are possible lengths of the shortest straw. The two lengths are 9 inches and 12 inches.

Answer 1

To find the missing side of a triangle you have to apply the Phytagorean Theorem:

a²+b²=c²

So b and c is given

b=9

c=12

a²+9²=12²

so you plug in the numbers inside the formula.

You should get the solution like this:

a²+81=144

-81 -81

__________

a²=63

Next you subtract 81 from both sides and then you square the difference to get the result of the missing side.

a²⁼63

sqrt(a)=sqrt(63)

Now you have to use prime factorization in order to square root 63 since it is not a perfect square.

The solution would turn out to be 3sqrt(7) since after prime factorization.

ANS: 3sqrt(7)

Answer 2

Recognizing the Pythagorean triple 3-4-5, the shortest length can be 9 (if the longest straw is greater than 15); if 12 is the longest straw, we know that it would be a right triangle if the shortest side was

√(12² - 9²) =

√63 =

3√7

Because the two longer sides are 9 and 12, the shortest side has to be longer than 12 - 9 = 3

Thus, it is obtuse if 3 < x < 3√7