14 Thiangle $$ K C M $$ is dilated by a scale factor of 3 to map onto triangle $$ D R S $$. Which statement is not always true? $$ N) \angle K=\angle D $$ (4) $$ K M=\frac{1}{3} D S $$ (3) The area of $$ riangle D R S $$ is 3 times the area of $$ riangle K L M $$. (4) The perimeter of $$ riangle D R S $$ is 3 times the perimeter of $$ riangle K L M $$.

Question

14 Thiangle $$ K C M $$ is dilated by a scale factor of 3 to map onto triangle $$ D R S $$. Which statement is not always true? $$ N) \angle K=\angle D $$ (4) $$ K M=\frac{1}{3} D S $$ (3) The area of $$ 	riangle D R S $$ is 3 times the area of $$ 	riangle K L M $$. (4) The perimeter of $$ 	riangle D R S $$ is 3 times the perimeter of $$ 	riangle K L M $$.

UpStudy AI · Accepted Answer

To analyze the statements regarding the dilation of triangle $$ KCM $$ to triangle $$ DRS $$ with a scale factor of 3, let's break down the properties of dilation and the implications for the triangles involved.

1. **Dilation Properties**:
   - When a triangle is dilated by a scale factor $$ k $$, the angles remain the same, meaning corresponding angles are equal.
   - The lengths of corresponding sides are multiplied by the scale factor $$ k $$.
   - The area of the dilated triangle is $$ k^2 $$ times the area of the original triangle.
   - The perimeter of the dilated triangle is $$ k $$ times the perimeter of the original triangle.

Now, let's evaluate each statement:

1. **Statement (1)**: $$ \angle K = \angle D $$
   - This statement is true because the angles remain unchanged during dilation.

2. **Statement (2)**: $$ KM = \frac{1}{3} DS $$
   - This statement is **not true**. Since $$ DS $$ is a side of triangle $$ DRS $$ and $$ KM $$ is a corresponding side of triangle $$ KCM $$, we have $$ KM = \frac{1}{3} DS $$ is incorrect. The correct relationship is $$ DS = 3 	imes KM $$.

3. **Statement (3)**: The area of triangle $$ DRS $$ is 3 times the area of triangle $$ KCM $$.
   - This statement is false. The area of triangle $$ DRS $$ is $$ 3^2 = 9 $$ times the area of triangle $$ KCM $$.

4. **Statement (4)**: The perimeter of triangle $$ DRS $$ is 3 times the perimeter of triangle $$ KCM $$.
   - This statement is true because the perimeter scales linearly with the scale factor.

### Conclusion:
The statement that is **not always true** is:
- **(2)** $$ KM = \frac{1}{3} DS $$ (This is incorrect; it should be $$ DS = 3 	imes KM $$).

Additionally, statement (3) is also incorrect regarding the area, but since the question asks for a single statement that is not always true, statement (2) is the most direct answer.
The statement that is not always true is: $$ KM = \frac{1}{3} DS $$.

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