Question

1. in the diagram, $\triangle jkl \sim \triangle mnp$.

a. maya and dylan each write a proportion to solve for the unknown side length $a$.
maya: $\frac{a}{24}=\frac{6.5}{6}$
dylan: $\frac{a}{6.5}=\frac{24}{6}$
whose proportion is correct? explain.
b. maya and dylan each write an equation to solve for the unknown side length $c$.
maya: $6^2 + c^2 = 6.5^2$
dylan: $\frac{c}{24}=\frac{6}{6.5}$
whose equation is correct? explain.

Explanation:

for part a:

Step1: Match corresponding sides

In $\triangle JKL \sim \triangle MNP$, $\overline{JK}$ (length 6.5) corresponds to $\overline{MN}$ (length $a$), and $\overline{JL}$ (length 6) corresponds to $\overline{MP}$ (length 24). The proportion of corresponding sides must be equal: $\frac{\text{Side of larger triangle}}{\text{Corresponding side of smaller triangle}} = \frac{a}{6.5} = \frac{24}{6}$.

Step2: Compare to given proportions

Dylan's proportion $\frac{a}{6.5} = \frac{24}{6}$ matches the correct correspondence of similar triangle sides. Maya's proportion mismatches the corresponding sides.

for part b:

Step1: Analyze Maya's equation

$\triangle JKL$ is a right triangle, so the Pythagorean theorem applies: $\text{leg}_1^2 + \text{leg}_2^2 = \text{hypotenuse}^2$. Here, $\overline{JL}=6$, $\overline{KL}=c$, $\overline{JK}=6.5$ (hypotenuse), so $6^2 + c^2 = 6.5^2$ is valid.

Step2: Analyze Dylan's proportion

Dylan's proportion $\frac{c}{24} = \frac{6}{6.5}$ mismatches corresponding sides: $c$ corresponds to $b$, not 24; 6 corresponds to 24, not 6.5.

Answer:

a. Dylan's proportion is correct. For similar triangles $\triangle JKL \sim \triangle MNP$, corresponding sides must be proportional: $\overline{MN}$ ($a$) corresponds to $\overline{JK}$ (6.5), and $\overline{MP}$ (24) corresponds to $\overline{JL}$ (6), so $\frac{a}{6.5} = \frac{24}{6}$ is the right proportion, which matches Dylan's work. Maya's proportion incorrectly pairs non-corresponding sides.
b. Maya's equation is correct. $\triangle JKL$ is a right triangle, so the Pythagorean theorem $6^2 + c^2 = 6.5^2$ correctly relates its legs and hypotenuse to solve for $c$. Dylan's proportion uses mismatched corresponding sides of the similar triangles, so it is incorrect.