Video Transcript
The lengths of a triangle are 12
centimeters, five centimeters, and 11 centimeters. Find the radius of the interior
circle touching the sides using the formula 𝑟 equals the area of the triangle
𝐴𝐵𝐶 over 𝑝, where 𝑝 is half of the triangle’s perimeter.
Well, to be able for us to use the
formula for the radius of the circle, what we need to do is find the area of the
triangle. And what we’re told are three side
lengths of our triangle: 12, five, and 11. Well, if we have three side lengths
of a triangle, then what we can use is Heron’s formula to find the area. And Heron’s formula tells us that
the area is equal to the square root of 𝑠 multiplied by 𝑠 minus 𝑎 multiplied by
𝑠 minus 𝑏 multiplied by 𝑠 minus 𝑐, where 𝑎, 𝑏, and 𝑐 are the side lengths of
our triangle. And 𝑠 is equal to 𝑎 plus 𝑏 plus
𝑐 over two because it’s the semiperimeter of our triangle. Well, actually, this ties in with
the formula we have for the radius because we’re told that the formula for the
radius is equal to the area of the triangle divided by 𝑝, where 𝑝 is half of the
triangle’s perimeter. Well, in fact, this is the same as
𝑠 because they both mean the semiperimeter or half of the triangle’s perimeter.
So, the first thing we want to do
is find out the area of the triangle. And to do that, firstly, we need to
find the semiperimeter or 𝑝, the half-perimeter of the triangle. Well, this is gonna be equal to 12
plus five plus 11 over two. So, this is gonna be equal to 14
centimeters. So great, what we can do now is
substitute this into Heron’s formula. And when we do that, we’re gonna
get the area is equal to the square root of 14 multiplied by 14 minus 12 multiplied
by 14 minus five multiplied by 14 minus 11, which is equal to root 756. And if we simplify this, we get six
root 21. And we’re gonna keep it in surd
form to keep accuracy, and the units for this will be centimeters squared because
it’s our area.
Okay, great. So, now, we have everything we need
to substitute into the formula to find the radius. What we’ve done is that we sketched
what we’re trying to find because what we’re trying to find is the radius of the
interior circle which touches the sides of our triangle. So, we can say that the radius is
equal to six root 21 over 14. And we know that because 14 was our
𝑠, our semiperimeter. And we’d already said that this is
the same as 𝑝. So, therefore, this is gonna give
us our final answer, which is the radius of the interior circle is three over seven
multiplied by root 21 centimeters.
