Video Transcript
The area of the triangle whose side lengths are three centimeters, six centimeters, and seven centimeters equals
something cm squared. So, in this question, what we’re
asked to do is find the area of the triangle. And what we’re given are the three
side lengths. And what we know is that if we have
the three side lengths and we want to find the area, what we can use is Heron’s
formula. And what Heron’s formula tells us
is that if we have triangle 𝑎𝑏𝑐, then 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
𝑠 represents the semiperimeter, which is half the perimeter of our triangle. And we’ve got a formula for that;
𝑠 is equal to 𝑎 plus 𝑏 plus 𝑐 over two.
Okay, great. So now we’ve remembered what
Heron’s formula is and what the formula for the semiperimeter is. We can use this to find the area of
our triangle. So first of all, we can find our
𝑠, our semiperimeter, cause this is equal to three plus six plus seven over two,
which is gonna be equal to 16 over two, which is eight. So, we know that the semiperimeter
would be eight centimeters. So now we’ve got the
semiperimeter. What we can do is plug this into
the Heron’s formula for the area. So, what we get is the area is
equal to the square root of eight multiplied by eight minus three multiplied by
eight minus six multiplied by eight minus seven. And because of the way that
formula’s set up, it doesn’t matter which one of our sides are 𝑎, 𝑏, or 𝑐, which
is gonna be equal to the square root of eight multiplied by five multiplied by two
multiplied by one, which is gonna give us root 80.
And then, what we’re going to do is
simplify our root 80. And what we’re going to get is root
16 multiplied by root five. And that’s because we apply one of
our radical or surd rules, which is that root 𝑎 multiplied by root 𝑏 is equal to
root 𝑎𝑏. And therefore, what we can say that
the area of the triangle whose side lengths are three centimeters, six centimeters,
and seven centimeters is four root five centimeters squared.
