An obtuse triangle has a side length of 16, and a side length of 21. The third side length is also a positive integer. How many different possible values are there for the third side length?

Question

I had gotten 37, but apparently that wasn't right.

Answer 1

In an acute triangle, all three internal angles are less than 90 degrees.

A right triangle has a 90° angle.

An obtuse triangle has one angle that is more than 90°

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For a triangle with sides of 16 and 21, the third side must be greater than 5 and less than 37.

A right triangle occurs when our numbers move from an obtuse triangle to an acute triangle OR from an acute triangle to an obtuse triangle. That happens when:

1) 21*21 = 16*16 + 13.6 * 13.6

2) 26.4 * 26.4 = 16*16 + 21*21

With a side length of 6,7,8,9,10,11,12,13 triangle is obtuse

With a side length of 14,15,16,17,18, 19, 20, 21, 22,23, 24, 25, 26 triangle is acute

With a side length of 27, 28, 29, 30, 31, 32, 33, 34, 35, 36 triangle is obtuse

TOTAL FOR OBTUSE: 18

Answer 2

Youhave to know the triangle inequality for the lengths of sides of triangles.

If two sides were 16 and 21 then the other has to be larger than 5 and smaller than 37. You can draw some sketches and you will see that if the third "side" were 5 or 37, you would not have a triangle.

The problem says out their side must be an integer so that leaves 6 to 36 but we are also told that the triangle is "obtuse." That means that many of those values, you will have an acute triangle so that you have to remove those from the list.

You can use the pythagorean theorem to see where the angle between the first two sides would be 90 degrees. Third side = sqrt(16^2 + 21^2) = 26.4. That tells is that if the third side is 26 or less, we'll have an acute angle (less than 90 degrees) and that if it is 27 or bigger, we'll have an obtuse angle (greater than 90 degrees). So the choices for the third side are 27 to 36, or 10 possibilities.

Answer 3

10 possibilities, all integers from 27 to 36

16+211=37 for the longest possible length of the 3rd side, but if actually 37, it collapses into a straight line with angles 0, 0 and 180. So slightly less than 37 is the largest possible for the third side.

To still be obtuse, it has to have the 3rd side such that it creates greater than a right triangle. So calculate the 3rd side for a right triangle, and make it slightly larger.

16 squared plus 21 squared = 3rd side squared.

256+441= 697 square root of 697 is 26.4

The 3rd side can be any integer between 26.4 and 37

Any 3rd side with less than 26.4 will create an acute, not obtuse triangle, with all angles less than 90 degrees.

any 3rd side with 37 or more will no longer be a triangle.

It can be 27, 28, 29, 30, 31, 32, 33, 34, 35, or 36

10 possibilities