Question

Three square mirrors are used for a light reflection experiment. The ratio of the side length of Mirror A to the side length of Mirror B is $5:6$. The ratio of the area of Mirror B to the area of Mirror C is $16:25$. The perimeter of Mirror C is $280$ centimeters. What is the area of Mirror A? Justify your answer.

Answer 1

Step 1:Find the length of mirror C.

Given, The ratio of the side length of Mirror A to the side length of Mirror B is $5:6$.

The ratio of the area of Mirror B to the area of Mirror C is $16:25$.

The perimeter of Mirror C is $280$ centimeters.

Since, we know that perimeter of square = $4\times$side length of square.

So, the side length of mirror C = $\frac{Perimeter}{4}=\frac{280}{4}=70\mathrm{cm}$.

Step 2: Find the length of mirror B.

Now, when two figures are similar then,

$\frac{AreaoffigureA}{AreaoffigureB}=\frac{{(SidelengthoffigureA)}^2}{{(SidelengthoffigureB)}^2}$

So, for Mirror C and Mirror B for similar figures,

$$\begin{gathered} \Rightarrow \frac{{(SidelengthoffigureB)}^2}{{(SidelengthoffigureC)}^2}=\frac{AreaofmirrorB}{AreaofmirrorC} \\ \Rightarrow \frac{{(SidelengthoffigureB)}^2}{{(SidelengthoffigureC)}^2}=\frac{16}{25} \\ \Rightarrow \frac{(SidelengthoffigureB)}{(SidelengthoffigureC)}=\frac{4}{5} \\ \Rightarrow SidelengthofmirrorB=\frac{4}{5}\times sidelengthofmirrorC \\ \Rightarrow SidelengthofmirrorB=\frac{4}{5}\times 70=56cm \end{gathered}$$

Step 3: Find the area of mirror A.

Since, we know that, area = side $\times$side.

So, area of Mirror B = $56\times 56=3136sq.cm$

Now, using the criteria for similar figures.

$$\begin{gathered} \Rightarrow \frac{AreaofmirrorA}{AreaofmirrorB}=\frac{{(sidelengthofmirrorA)}^2}{{(sidelengthofmirrorB)}^2} \\ \Rightarrow \frac{AreaofmirrorA}{3136}=\frac{5^2}{6^2} \\ \Rightarrow AreaofmirrorA=2177.7sq.cm \end{gathered}$$

Step 4: Final answer.

Hence, the area of mirror A is $2177.7sq.cm$.