Question

determine the validity of the conditional statement below and the validity of its converse. if possible, re-write the given statement as a biconditional.

if a circle has a radius of 3 meters, then it has a circumference of 6π meters.

show your work here

conditional statement is:

○ true ○ false

converse of statement is:

○ true ○ false

biconditional form:

○ if a circle has a circumference of 6π meters, then it has a radius of 3 meters.

○ a circle has a radius of 3 meters if and only if it does not have a circumference of 6π meters.

○ a circle has a radius of 3 meters if and only if it has a circumference of 6π meters.

○ biconditional is not possible.

Explanation:

Step1: Analyze Conditional Statement

The formula for the circumference \( C \) of a circle is \( C = 2\pi r \), where \( r \) is the radius. For \( r = 3 \) meters, \( C = 2\pi(3)=6\pi \) meters. So the conditional "If a circle has a radius of 3 meters, then it has a circumference of \( 6\pi \) meters" is True.

Step2: Analyze Converse Statement

The converse is "If a circle has a circumference of \( 6\pi \) meters, then it has a radius of 3 meters". Using \( C = 2\pi r \), solve for \( r \): \( r=\frac{C}{2\pi} \). Substituting \( C = 6\pi \), we get \( r=\frac{6\pi}{2\pi}=3 \) meters. So the converse is True.

Step3: Analyze Biconditional Form

A biconditional is "A if and only if B" when both "If A, then B" and "If B, then A" are true. Since both the conditional and its converse are true, the biconditional is "A circle has a radius of 3 meters if and only if it has a circumference of \( 6\pi \) meters".

Answer:

Conditional statement: True
Converse of statement: True
Biconditional form: A circle has a radius of 3 meters if and only if it has a circumference of \( 6\pi \) meters.