Using the figure below, find the ratio between the area of the parallelogram 𝑋𝑌𝑍𝐿 and the area of the triangle 𝐴𝐵𝐶 in its simplest form.
So the first thing we’re going to do to solve the problem is find the area of our parallelogram. Well, the area of parallelogram is equal to the base multiplied by the height. The height must be the perpendicular height. So if we look at our figure that we have 𝑋𝑌𝑍𝐿, it’s gonna be 12 multiplied by six, not 10. That’s the slanted edge that’s often put in there to try and catch people out. And sometimes it’s a common mistake to use that instead of the six. So therefore, what we can say is that the area is gonna be equal to 72 centimeters squared. And that’s because 12 multiplied by six is 72. So great, that’s the area of our parallelogram.
Now we’re gonna move on and find the area of our triangle. Well, to find the area of our triangle 𝐴𝐵𝐶, what we’re going to do is use the area of a triangle formula, and that is a half base times the height, so the area is equal to a half multiplied by the base multiplied by the height. So therefore, for our triangle 𝐴𝐵𝐶, it’s gonna be a half multiplied by 12 multiplied by five, which is gonna be equal to 30 centimeters squared. Okay, great. So now what do we need to do? Well, the question asks us to find the ratio between the areas of our parallelogram and our triangle.
Well, because we’re dealing with a ratio, we don’t have to worry about the units, which was centimeters squared. So that means the ratio between the areas is going to be 72 to 30, remembering that we put the 72 first because the ratio that we want is parallelogram to triangle. So we put the area of the parallelogram first. So have we finished? Have we solved the problem? Well, yes, we have found the ratio between the areas. However, it is not in its simplest form. So what we need to do now is simplify so that it gets into its simplest form.
Well, we know that six is a common factor of both 72 and 30. So what we’re gonna do is divide both side of our ratio by six. And when we do that, we get 12 and five. So therefore, we can say that the ratio between the area of our parallelogram and our triangle is 12 to five, and that’s in its simplest form.