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All courses Geometry · G-C.1 14 of 57
Prove that all circles are similar.

Identify two separate circle center-radius records

Problem
Identify \(\text{both}~\text{circle}~\text{records}\) from Circle \(A\) has named center \(A\) and radius \(3\); Circle \(B\) has named center \(B\) and radius \(9\).
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Hint

Create one row for Circle A and one for Circle B, then copy each stated center and radius into its matching row.

When each circle’s center and radius are stated, keep the two records separate and compute the requested ratio as rB/rA.

Solution walkthrough

01

Use the definition of a circle record

\[\text{record}=(\text{named}~\text{center},~\text{radius})\]

A circle is identified by its center and the constant distance from that center to every boundary point.

02

Record Circle A

\[\text{center}~A;~\text{radius}~3\]

The source statement names A as the center and gives a three-unit radial segment.

03

Record Circle B and compare

\[\begin{aligned} \text{center}~B;~\text{radius}~9 \\ 9/3=3 \end{aligned}\]

The second circle has named center B and radius 9. Its radius is three times Circle A's, which agrees with the inspected diagram.

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Another way

  1. Write the data as ordered records (A,3) and (B,9).

!

Common mistake

Do not treat the center names A and B as radius labels. Each letter names a point; the numerical distance is the radius.

Answer circle diagram: Circle A has center A and radius 3; Circle B has center B and radius 9; radius ratio rB/rA=3.
Circle A source=stated; Circle A center=A; Circle A radius=3; Circle B source=stated; Circle B center=B; Circle B radius=9; radius ratio rB/rA=3

Solution walkthrough video