Question

pentagon m is a scaled copy of pentagon l.
pentagon l has a side labeled 3\\(\frac{1}{3}\\), a side labeled 5, and a base labeled 1\\(\frac{1}{2}\\). pentagon m has a side labeled 2\\(\frac{1}{2}\\), a side labeled n, and a base labeled 1\\(\frac{1}{8}\\).
what is the value of n?

Explanation:

Step1: Find scale factor

Scale factor = $\frac{\text{Side of M}}{\text{Corresponding side of L}} = \frac{2\frac{1}{2}}{3\frac{1}{3}} = \frac{\frac{5}{2}}{\frac{10}{3}} = \frac{5}{2} \times \frac{3}{10} = \frac{3}{4}$

Step2: Calculate n

$n = 5 \times \frac{3}{4} = \frac{15}{4}$? Wait, no—wait corresponding sides: wait the base of L is $1\frac{1}{2} = \frac{3}{2}$, base of M is $1\frac{1}{8} = \frac{9}{8}$. Scale factor = $\frac{9/8}{3/2} = \frac{9}{8} \times \frac{2}{3} = \frac{3}{4}$. Correct. Then the side 5 in L corresponds to n in M: $n = 5 \times \frac{3}{4} = \frac{15}{4}$? Wait no, wait the other side: M's side is $2\frac{1}{2} = 5/2$, L's is $3\frac{1}{3}=10/3$. Scale factor is (5/2)/(10/3)= 3/4. So yes, scale factor is 3/4. Then n = 5(3/4)=15/4? Wait wait the problem says pentagon M is scaled copy of L, so scale factor is M/L. So if L's side is 5, M's side n = 5(scale factor). Scale factor from L to M is 3/4, so n=15/4=3.75. Wait but let's check again: 3 1/3 is 10/3, 2 1/2 is 5/2. 5/2 divided by 10/3 is 3/4. Correct. 1 1/2 is 3/2, 1 1/8 is9/8. 9/8 divided by3/2 is3/4. Correct. So 5*(3/4)=15/4=3.75. Oh I made a mistake earlier. So the answer is 15/4 or 3.75.

Wait wait, no—wait the question says pentagon M is a scaled copy of pentagon L, so M = scale L. So scale factor k = M/L. So k = (2 1/2)/(3 1/3) = (5/2)/(10/3) = 3/4. So n = k 5 = (3/4)*5 =15/4=3.75. Yes, that's right.

Step1: Compute scale factor

Scale factor $k = \frac{2\frac{1}{2}}{3\frac{1}{3}} = \frac{5/2}{10/3} = \frac{3}{4}$

Step2: Find n

$n = 5 \times \frac{3}{4} = \frac{15}{4}$

Answer:

$\frac{10}{3}$