Question Video: Finding the Area of an Isosceles Triangle Mathematics

Calculate the area of the triangle 𝐴𝐵𝐶.


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.

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