Triangle APQ is the image of ABC under a dilation centered at vertex A with scale factor 1/2 Triangle RBT is the image of ABC under a dilation centered at vertex B with Scale factor 3/4 which statement about ABC APQ and RBT is correct
their areas are in the ratio 2:3:4
To determine which statement about triangles ABC, APQ, and RBT is correct, we need to analyze the effects of the dilations on each triangle.
First, let's consider triangle APQ, which is the image of triangle ABC under a dilation centered at vertex A with a scale factor of 1/2. This means that every corresponding side and angle of triangle APQ is half the length or measure of its corresponding side or angle in triangle ABC.
Next, let's analyze triangle RBT, which is the image of triangle ABC under a dilation centered at vertex B with a scale factor of 3/4. This implies that every corresponding side and angle of triangle RBT is three-fourths the length or measure of its corresponding side or angle in triangle ABC.
Now let's consider the given statements about the three triangles:
1. Triangle APQ is congruent to triangle ABC: This statement is incorrect because the scale factor for triangle APQ is 1/2, indicating that APQ is a smaller version of ABC, not congruent to it.
2. Triangle ABC is smaller than triangle RBT: This statement is incorrect because the scale factor for triangle RBT is 3/4, meaning RBT is smaller than ABC.
3. Triangle ABC and triangle RBT are similar: This statement is correct. Two triangles are similar when their corresponding angles are congruent and the corresponding sides are proportional. In this case, both triangle ABC and triangle RBT have corresponding angles that are congruent because both dilations preserve the angles. Additionally, the corresponding sides are proportional as the scale factor for triangle RBT is 3/4, which means that each side of RBT is three-fourths the length of the corresponding side in ABC. Thus, we can conclude that triangle ABC and triangle RBT are similar.
Therefore, the correct statement is: "Triangle ABC and triangle RBT are similar."