Question Video: Finding the Radius of the Interior Circle That Touches the Sides of a Triangle Using a Given Relation and the Triangle’s Side Lengths Mathematics

The lengths of a triangle are 12 cm, 5 cm, and 11 cm. Find the radius of the interior circle touching the sides using the formula 𝑟 = Area (△𝐴𝐵𝐶)/𝑝, where 𝑝 is half of the triangle’s perimeter.

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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.

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