Question

\\(\delta klm\\) is dilated by a scale factor of 3 to map onto \\(\delta drs\\). which statement is not true?

select one:
a. \\(km = \frac{1}{3} ds\\)
b. \\(kl = 3(dr)\\)
c. \\(\angle k \cong \angle d\\)
d. \\(m\angle s = m\angle m\\)

Explanation:

🆕 New Concept Discovered: Dilations and Similarity
How shapes change size but keep their angles and proportions.

Step 1: Understand the dilation mapping

When \(\Delta KLM\) is dilated to map onto \(\Delta DRS\), the corresponding vertices and sides match up in order:

• Vertex \(K\) corresponds to vertex \(D\)
• Vertex \(L\) corresponds to vertex \(R\)
• Vertex \(M\) corresponds to vertex \(S\)

This means \(\Delta KLM \sim \Delta DRS\) (the triangles are similar).

Step 2: Analyze the properties of dilation

A dilation by a scale factor of \(k = 3\) has two key properties:

1. Angles stay the same: Corresponding angles are congruent.

• \(\angle K \cong \angle D\) (Statement c is true)
• \(m\angle M = m\angle S\) (Statement d is true)

1. Sides change proportionally: Every side length of the new triangle (\(\Delta DRS\)) is \(3\) times the length of the corresponding side of the original triangle (\(\Delta KLM\)).

• \(DR = 3 \cdot KL\)
• \(RS = 3 \cdot LM\)
• \(DS = 3 \cdot KM\)

Step 3: Evaluate the side-length statements

Let's look at the remaining statements:

• Statement a: \(KM = \frac{1}{3} DS\)

Since \(DS = 3 \cdot KM\), dividing both sides by \(3\) gives \(KM = \frac{1}{3} DS\). This statement is true.

• Statement b: \(KL = 3(DR)\)

Since \(\Delta DRS\) is the larger image, the relationship is \(DR = 3(KL)\). Rearranging this gives \(KL = \frac{1}{3}(DR)\). Therefore, \(KL = 3(DR)\) is false.

Answer:

b. \(KL = 3(DR)\)