Given that triangle 𝐴𝐵𝐶 is similar to triangle 𝑃𝑄𝑅, if 𝐴𝐵 divided by 𝑃𝑄 is equal to one divided by three, find the area of triangle 𝐴𝐵𝐶 divided by the area of triangle 𝑃𝑄𝑅.
So for this question, the first thing to note is this little symbol here. This has been used to show us that triangle 𝐴𝐵𝐶 and 𝑃𝑄𝑅 are similar. Once we know that two triangles are similar, we know that their corresponding sides scale to the same proportion and that their corresponding angles are equal.
Okay, now that we understand this, we can draw our sketch of two similar triangles. We’ve labelled the large triangle 𝑃𝑄𝑅 and the small triangle 𝐴𝐵𝐶. To understand why, let’s look at the information given by the question.
The question tells us that length 𝐴𝐵 divided by length 𝑃𝑄 is equal to a third. It might be easier to see what this equation is telling us if we first multiplied both sides by 𝑃𝑄 and then we multiplied both sides by three to find that three times the length 𝐴𝐵 is equal to the length 𝑃𝑄.
Looking at our diagram, this must mean that three times the length 𝐴𝐵 is equal to the length 𝑃𝑄. In fact, here the question has given us what is called a scale factor, specifically, a scale factor for length. We can understand scale factors by imagining a transformation from one triangle to another.
The length scale factor — labelled 𝑆𝐹𝐿 here — can be found by taking a length on your final triangle and dividing it by the corresponding length on your initial triangle. In our case, the length 𝑃𝑄 is our initial length and the length 𝐴𝐵 is our corresponding final length.
To solve this question, we can use another scale factor: the area scale factor, which we’ll label 𝑆𝐹𝐴. The area scale factor can be found by dividing the area of your final triangle by the area of your initial triangle.
Looking at our diagram, we can see that the area of our initial triangle is the area of triangle 𝑃𝑄𝑅 and the area of our final triangle is the area of triangle 𝐴𝐵𝐶. Now, length and area scale factors relate together in the following way: if we say the length scale factor is equal to some constant 𝑘, then the area scale factor will be equal to the same constant 𝑘 squared.
We can, therefore, find the area scale factor by taking the length scale factor and squaring it. Since the question has given us a value for the length scale factor, we now have all the information we need. The area scale factor is the length scale factor or one over three squared.
When squaring a fraction, we can take the square of the top and the bottom half separately. One squared is one and three squared is nine. Since we know that the area scale factor is equal to the area of triangle 𝐴𝐵𝐶 divided by the area of triangle 𝑃𝑄𝑅, our answer is, therefore, one over nine.