Video Transcript
Calculate the area of the triangle
𝐴𝐵𝐶.
The first thing we might note when
we’re looking at this triangle is that it’s an isosceles triangle. We can tell this from the hash
notation indicating that we have two sides of equal length, which therefore fits the
definition of an isosceles triangle. In order to find the area of a
triangle, we’ll need the formula that the area of a triangle is equal to half
multiplied by the base multiplied by the perpendicular height.
If we were to try and immediately
calculate this however, we’d have a problem. This value of 10 centimeters
represents the slant height of the triangle but not the perpendicular height. The perpendicular height would look
like this. We can even define it with the
letter ℎ if we wish. As we’ve created two right
triangles here, we might consider the Pythagorean theorem, which tells us that the
square on the hypotenuse is equal to the sum of the squares on the other two
sides.
So let’s consider this triangle on
the left. We can say that this line from 𝐴
meets the line 𝐵𝐶 at point 𝑦. In order to use the Pythagorean
theorem, we need to know the length of this line segment 𝐵𝑦. Now, you might think that it’s very
clear that it’s six centimeters. But how can we be absolutely sure
that it is six centimeters?
Let’s consider the two
triangles. We’ve got triangle 𝐴𝐵𝑦 on the
left and triangle 𝐴𝐶𝑦 on the right. If we consider a pair of sides, we
know that side 𝐴𝐵 is equal to side 𝐴𝐶. Side 𝐴𝑦 is common to both
triangles. And finally, angle 𝐴𝑦𝐵 is equal
to angle 𝐴𝑦𝐶. They’re both 90 degrees. We can say then that triangle
𝐴𝐵𝑦 is congruent with triangle 𝐴𝐶𝑦 by using the right angle hypotenuse side
congruency criterion.
You might not need to show that
level of working in every question. But it’s good to demonstrate that
it means that this length of 𝐵𝑦 is the same as the length of 𝑦𝐶. They’ll both be six
centimeters. This working also proves an
important property of isosceles triangles that the median to the base of an
isosceles triangle is perpendicular to the base. In other words, this line from 𝐴
to 𝐵𝐶 connecting at the midpoint 𝑦 will be perpendicular to the base 𝐵𝐶.
Let’s continue with this question
and apply the Pythagorean theorem. Using the triangle 𝐴𝐵𝑦, we can
see that there’s a hypotenuse of 10 and the other two sides will be six and ℎ. So we write 10 squared is equal to
six squared plus ℎ squared. Evaluating the squares, 100 is
equal to 36 plus ℎ squared. Subtracting 36 from both sides
gives us 64 is equal to ℎ squared. We should recognize that 64 is a
perfect square. So when we take the square root,
we’ll have ℎ is equal to eight. And the units will be the length
units of centimeters.
In this question, remember that
we’re finding the area, not just the perpendicular height. So we’ll use the area formula
now. When we’re filling in our values
for the base and the height, remember that we’re using this base of 12 centimeters,
not six centimeters, and we’re multiplying that by a half and then by eight, the
perpendicular height. We can simplify before we multiply
to give us a value of 48. And as we’re working with an area,
we’ll need square units. We can give our answer then that
the area of triangle 𝐴𝐵𝐶 is 48 square centimeters.
