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 \( \triangle D R S \) is 3 times the area of \( \triangle K L M \). (4) The perimeter of \( \triangle D R S \) is 3 times the perimeter of \( \triangle K L M \).
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When a triangle is dilated by a scale factor, the relationships between angles, sides, area, and perimeter can change significantly. One common misconception is regarding the areas of the triangles involved. While the angles remain the same (angles in similar triangles are equal), and perimeters scale linearly with the scale factor, the area’s relationship is based on the square of the scale factor. Therefore, if \( K C M \) is dilated by a scale factor of 3, the area of \( \triangle D R S \) will actually be \( 3^2 = 9 \) times that of \( \triangle K C M \), not just 3 times. So to answer your question, the statement “The area of \( \triangle D R S \) is 3 times the area of \( \triangle K C M \)” is not true!
