Question

Right triangle ABC has inradius 1 and Sin A = 12/13. Find the length of the hypotenuse of ABC

Answer 1

6.5

Explanation:

You want the length of the hypotenuse of right triangle ABC with sin(A) = 12/13 and an inradius of 1.

Right triangle

The ratio of long leg to hypotenuse of 12/13 tells us this is a triangle with sides in the ratios 5 : 12 : 13. For the purpose of determining an inradius, we can assume these are the actual side lengths.

Inradius

The inradius of a triangle is ...

`r=\sqrt{((s-a)(s-b)(s-c))/(s)}\qquad\text{where }s=(a+b+c)/(2)`

For side lengths 5, 12, 13, we have ...

s = (5+12+13)/2 = 15

r = √((15 -5)(15 -12)(15 -13)/15) = 2

This tells us our triangle with sides 5, 12, 13 is 2 times the size of the one we want.

Hypotenuse

The length of the hypotenuse of ∆ABC is 13/2 = 6.5 units.

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Additional comment

We can use the Pythagorean theorem to find the length of the third side, given a side of 12 and a hypotenuse of 13.

a² +b² = c²

a² +12² = 13²

a = √(169 -144) = √25 = 5

It is easier to consult our memory of Pythagorean triples. The ones most commonly seen in algebra, trig, and geometry problems are ...

{3, 4, 5}, {5, 12, 13}, {7, 24, 25}, {8, 15, 17}, {9, 40, 41}

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