Find the similarity ratio of two circles with areas 75πcm2and27πcm2.

Question

sovienesY · Accepted Answer

Step 1: Take note of the details provided.
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: Determine the ratio of the two circles' surface areas.
$A_{1} : A_{2} = 75 π : 27 π$
$π$ 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

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

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