Question

The longest side of an isosceles obtuse triangle measures 20 centimeters. The other two side lengths are congruent but unknown. What is the greatest possible whole-number value of the congruent side lengths? 9 cm 10 cm 14 cm 15 cm

Answer 1

The greatest possible whole number length of these two sides is something that must be greater than 10 (so that it is greater then 20, because otherwise it wouldn't fit the triangle inequality property). That leaves us with two answers, 14 and 15. In addition, in an obtuse triangle, the square of the longest side is greater than the square of the two other sides. So, the sides must have a length L² that is less than 400. 15²·2 is 450, so the greatest possible value that works is 14. 

Answer 2

The two side lengths are congruent, which means that they will have the same length.

The first step of this equation concerns the Triangle Inequality Theorem, which states that any two sides of a triangle must be greater than or equal to the third, last side.
The longest side is 20cm, which means that the missing side lengths must add up to a value greater than 20. We can write the inequality:

$2s \ \textgreater \ 20$

s represents the missing side lengths that we must find. The coefficient is 2, because the side lengths are identical to each other.

We can simplify the inequality by dividing both sides by 2:

$s \ \textgreater \ 10$

This means that the side lengths cannot be 9 or 10cm.

The second part of this question involves the Pythagorean Theorem, but tailored to fit inequalities. This modification is aptly named the Pythagorean Inequality.

For obtuse triangles, the theorem uses the following formula:

$a^2 + b^2 \ \textless \ c^2$

The biggest side length's square must be greater than the squares of the other two sides to keep the triangle obtuse.

We can plug in 14 and 15 to this formula to see if the expression stays true:

$14^2 + 14^2 \ \textless \ 20^2$
$196 + 196 = 392 \ \textless \ 400$

$15^2 + 15^2 \ \textless \ 20^2$
$225 + 225 = 450 \ \textgreater \ 400$

Plugging in 15 into the equation makes the expression false.

The greatest possible whole-number value of the congruent side lengths is 14cm.