The triangles in the figure are all congruent. Determine the area of the figure.
Two shapes are congruent if they’re exactly the same shape and size. This means that if we drew the shapes on a piece of paper, we could cut one of them out and place it directly on top of the other although we may need to turn it round in order to do so. Looking at the diagram, we can see that it consists of five triangles, which we’re told are all congruent. So in order to work out the area of the figure, we just need to work out the area of one triangle. And we can then multiply it by five.
Let’s consider just one of these triangles then, the one that I’ve shaded in pink. We know that, to find the area of a triangle, we can use the formula, a half multiplied by base multiplied by the perpendicular height. We can see that we’ve been given the base of this triangle. It’s 86 meters. But what about the perpendicular height?
Well, there’re actually two ways that we could work this out. Firstly, we notice that, in the centre of our diagram, there’re actually three lots of this perpendicular height stacked on top of one another. And they’re all the same as the triangles are congruent. We also know that the total height of the figure is 129 meters. So we can form an equation. Three ℎ equals 129. By dividing both sides of this equation by three, we find that the perpendicular height of our triangles is 43 meters. Another way to see this is that, for the triangle on the right of the diagram, its base is actually its vertical side. So this length here is also 86 metres.
We can therefore work out the height of the top congruent triangle by subtracting our value of 86 meters from the total height of the figure, 129 meters. 129 minus 86 is 43. So again, we found that the perpendicular height of our triangles is 43 meters. Substituting the values for the base and perpendicular height into our formula for the area of our triangle then, we have that the area of each triangle is a half multiplied by 86 multiplied by 43.
Remember though that there’re five of them. So to work out the area of the complete figure, we also need to multiply by five. Evaluating this on a calculator gives 9245. The units for this area will be metres squared because the lengths for the figure were given in metres.
There are other approaches we could’ve taken to working out the area once we’ve worked out the measurements. For example, we could’ve worked out the area of the square containing four of the congruent triangles by multiplying 86 by 86 and then worked out the area of the remaining triangle using a half base times height. But I think working out the area of one triangle and then multiplying it by five is the most straightforward way.