Show that the two triangles are similar.
When two triangles are similar, one will be an enlargement of the other. They should have the same interior angles. Since we have not been given the size of any of the angles in these triangles, we need to use the fact that if they are similar, one will be an enlargement of the other.
When a shape is enlarged, we use a scale factor. This tells us how much larger one shape is than the other. And we can use this formula to find the scale factor. Scale factor is equal to new divided by old, where new and old are corresponding new measurements and old measurements in our triangle.
In this case, for example, we could call five-thirds the new measurement and four the corresponding old measurement. The scale factor here would be found by dividing five-thirds by four. To divide by four, we write four as four over one. We then change the division to a multiplication, and we find the reciprocal of the second number. The reciprocal of four over one is one over four, or one-quarter. We then multiply the numerator of the first fraction by the numerator of the second to get five, and the two denominators: three multiplied by four is 12.
We can see that the scale factor for this first pair of sides is five twelfths. If we can show that the other two sides have been enlarged by the same scale factor, then we will know the triangle has been enlarged and therefore they must be similar.
Let’s now look at the sides measuring fifteen quarters. We can see that the corresponding old length is nine centimeters. When we divide the new length by its corresponding old length, we get fifteen quarters divided by nine. Remember, we can write nine as nine over one. And once again, we can change the division to a multiplication, and we find the reciprocal of nine over one, which is one-ninth. Multiplying the numerators we get 15, and the denominators we get 36.
We do, however, need to simplify this fraction. Both 15 and 36 are divisible by three. 15 divided by three is five, and 36 divided by three is 12. We’ve shown that the scale factor for these two corresponding sides is five twelfths. Let’s add a little key so we can follow what’s happening.
The final two corresponding sides are six, which has been enlarged to five over two. Dividing the new length by its corresponding old length gives us five over two divided by six or five over two divided by six over one, which we now know is the same as five over two multiplied by one-sixth, since one-sixth is the reciprocal of six over one. Five multiplied by one is five, and two multiplied by six is 12.
Once again, we’ve shown that the scale factor for this final pair of sides is five twelfths. We can see then that the first triangle has been enlarged consistently by a scale factor of five twelfths. This means that the two triangles are similar.