Find the similarity ratio of two circles with areas 75 pi cm^2 and 27 pi cm^2.

Question

Find the similarity ratio of two circles with areas

75 π c m^{2} and 27 π c m^{2}

.

Answer 1

Step 1: Note down the given information
Area of the first circle,

A_{1} = 75 π c m^{2}

Area of the second circle,

A_{2} = 27 π c m^{2}

Let the radius of the circles be

r_{1} and r_{2}

respectively.
Step 2: Calculate the ratio of areas of the the two circles

A_{1} : A_{2} = 75 π : 27 π

π

cancels pi and we have both 27 and 75 divisible by 3
So, we can simplify this as

A_{1} : A_{2} = 25 : 9

....(1)
Step 3: Calculate the ratio of areas in terms of

r_{1} and r_{2}

The scale (similarity) factor for circles is the ratio of the radius
Similarity factor =

r_{1} : r_{2}

Area of a circle =

π r_{2}

so

A_{1} : A_{2} = π r_{1}^{2} : π r_{2}^{2}

This simplifies to

A_{1} : A_{2} = r_{1}^{2} : r_{2}^{2}

.....(2)
Step 4: Use equation 1 and 2 to deduce similarity ratio
From 1 and 2, we have

r_{1}^{2} : r_{2}^{2} = 25 : 9

Taking square root both sides we get

r_{1} : r_{2} = 5 : 3

Result: So, the similarity ratio is 5 : 3

Expert Answer