Question

1. the lengths of the sides of a pentagon are 2\, 6\, 10\, 14\, and 24\. calculate the lengths of the sides of a similar pentagon if the shortest side is 5\. draw a diagram

Explanation:

Step1: Identify the scale factor

The shortest side of the original pentagon is 2", and the shortest side of the smaller pentagon is 5". Wait, no—wait, the original sides are 2", 6", 10", 14", 24"? Wait, no, maybe the original shortest side is 2", and the smaller pentagon's shortest side is 5"? Wait, no, the problem says "the lengths of the sides of a pentagon are 2", 6", 10", 14", and 24". Calculate the lengths of the sides of a smaller pentagon if the shortest side is 5". So first, find the scale factor. The original shortest side is 2", the new shortest side is 5". So scale factor \( k = \frac{5}{2} \).

Step2: Scale each side

Now, multiply each original side length by the scale factor \( k = \frac{5}{2} \).

• For 2": \( 2 \times \frac{5}{2} = 5 \) (which is the given shortest side, checks out)
• For 6": \( 6 \times \frac{5}{2} = 15 \)
• For 10": \( 10 \times \frac{5}{2} = 25 \)
• For 14": \( 14 \times \frac{5}{2} = 35 \)
• For 24": \( 24 \times \frac{5}{2} = 60 \)

Answer:

The side lengths of the smaller pentagon are 5", 15", 25", 35", and 60".