Question

1. a small square of area 5 is inscribed in a large square, creating four identical triangles, as shown below. the lengths of the two legs of each triangle are 1 and 2. what is the ratio of the side length of the large square to the side length of the small square? (adapted from 2023 amc 10 problem, question #11)

a. (\frac{4}{sqrt{5}}) b. (\frac{3}{2}) c. (\frac{3sqrt{5}}{5}) d. (3 - sqrt{5}) e. (sqrt{5}-1)

Explanation:

Step1: Calculate the side length of the small square

Given the area of the small square \(A = 5\). Using the formula for the area of a square \(A=s^{2}\) (where \(s\) is the side - length), we have \(s_{small}=\sqrt{5}\) (since \(s^{2}=5\) implies \(s = \sqrt{5}\)).

Step2: Calculate the side length of the large square

The side - length of the large square \(s_{large}\) can be found by using the fact that if the legs of the right - triangle are \(a = 1\) and \(b = 2\), then \(s_{large}=1 + 2=3\) (by observing the composition of the large square's side from the sum of the legs of the right - triangle).

Step3: Calculate the ratio

The ratio of the side - length of the large square to the side - length of the small square is \(\frac{s_{large}}{s_{small}}\). Substitute \(s_{large}=3\) and \(s_{small}=\sqrt{5}\) into the ratio formula. We get \(\frac{3}{\sqrt{5}}=\frac{3\sqrt{5}}{5}\) (rationalizing the denominator: \(\frac{3}{\sqrt{5}}\times\frac{\sqrt{5}}{\sqrt{5}}=\frac{3\sqrt{5}}{5}\)).

Answer:

\(\frac{3\sqrt{5}}{5}\) (Option C)