Triangle APQ is the image of ABC under a dilation centered at vertex A with scale factor ½. Triangle RBT is the image of ABC under a dilation centered at vertex B with scale factor ¾ . Which statement about ABC , APQ , and RBT is correct? none of the triangle are similar
All three triangles are similar
Triangle APQ an RBT are not similar
scaling preserves angles, so all three triangles are similar.
Well, it seems we have a situation here! Based on the information given, it is safe to say that the statement "All three triangles are similar" is correct. We have two dilations going on here, one centered at vertex A with a scale factor of 1/2, and another centered at vertex B with a scale factor of 3/4. Since any dilation results in similar triangles, all three triangles - ABC, APQ, and RBT - will indeed be similar. So, let's give them a round of applause for being similar triangles! 🎉👏
All three triangles are similar.
To determine if the triangles are similar, we need to check if their corresponding angles are congruent and their corresponding side lengths are proportional.
Let's start by examining the angles:
- Triangle APQ is the image of ABC under a dilation from vertex A. This means that angle APQ is equal to angle BAC, and angle PAQ is equal to angle ABC. Similarly, angle AQP is equal to angle ACB.
- Triangle RBT is the image of ABC under a dilation from vertex B. This means that angle RBT is equal to angle BAC, and angle TRB is equal to angle ABC. Similarly, angle BRT is equal to angle ACB.
Since the corresponding angles of both triangles are congruent, we move on to comparing side lengths:
- The dilation centered at vertex A with a scale factor of 1/2 means that the corresponding side lengths of APQ and ABC are multiplied by 1/2.
- The dilation centered at vertex B with a scale factor of 3/4 means that the corresponding side lengths of RBT and ABC are multiplied by 3/4.
By comparing the corresponding side lengths, we see that:
- The side lengths of APQ are half the lengths of ABC.
- The side lengths of RBT are three-fourths the lengths of ABC.
Therefore, the side lengths of APQ and RBT are not proportional to each other. Hence, the correct statement is:
Triangle APQ and RBT are not similar.