A triangle 𝐴𝐵𝐶 has been dilated from a center 𝑃 by a scale factor of three to triangle 𝐴 prime 𝐵 prime 𝐶 prime. Are triangles 𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime similar? Are triangles 𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime congruent?
In this question, we’re told that there’s a triangle 𝐴𝐵𝐶. We don’t know what it looks like, but we’re told that there’s a center 𝑃 and that 𝐴𝐵𝐶 has been dilated by a scale factor of three about this center. So let’s see if we can model this triangle. Let’s take some squared paper and draw any triangle 𝐴𝐵𝐶. On this triangle, the base is two units and the height is three units. We can draw the point 𝑃 anywhere as we’re not told where it is. So let’s draw it here on the left side of the triangle.
When we perform a dilation, we take each vertex in turn and look at it in relation to the center. The point 𝐴 is two units right from the center. So the image 𝐴 prime will be three times that distance from the center, which means it will be six units to the right. 𝐵 prime is two units to the right and three units down from the center. Therefore, 𝐵 prime the image will be three times that distance from the center. In the same way, we can find the image of 𝐶 which will be 𝐶 prime.
Let’s take a look at the sides of this new triangle 𝐴 prime 𝐵 prime 𝐶 prime. Even without drawing this new triangle, we should remember that when there’s been a dilation of scale factor three, then each of the new lengths will be three times of the original lengths. When we have a dilation, the corresponding pairs of angles will all stay the same size.
Let’s think about the first question which asks if these two triangles are similar. Informally, we can say that similar means the same shape. A better mathematical definition is that similar shapes will have corresponding angles equal, and corresponding sides are in proportion. So if we look at our two triangles, we can already see that the corresponding angles are equal. But are the sides in proportion? Well, yes they are because we know that the triangle 𝐴 prime 𝐵 prime 𝐶 prime will all have sides which are three times as long as in 𝐴𝐵𝐶.
We can then answer the first part of this question that yes, these two triangles would be similar.
In the second part of this question, we’re asked if these triangles are congruent. Informally, we can say that congruent triangles will be the same shape and the same size. Our mathematical definition for congruency is that corresponding pairs of angles are equal and corresponding sides are equal. We know that we have corresponding angles equal in our two triangles. But are the sides equal? Well, no they’re not because we’ve already shown that triangle 𝐴 prime 𝐵 prime 𝐶 prime has sides that are three times as long as those in 𝐴𝐵𝐶.
Therefore, the answer for the second part of this question is no, as these two triangles are not congruent.