Question

In a geometry class, the students were asked to write statements that are logically equivalent to the statement shown below.

If a line segment joins the center of a circle with a point on the circle, then it is a radius.

Below are the responses of four students. Which student’s response is correct?

Answer 1

Student 2: If a line segment is not a radius, then it joins the center of a circle with a point on the circle.

Student 1: If a line segment is not a radius, then it does not join the center of a circle with a point on the circle.

Student 3: If a line segment does not join the center of a circle with a point on the circle, then the line segment is a radius.

Student 4: If a line segment joins the center of a circle with a point on the circle, then the line segment is not a radius.

Answer 2

The statement that is logically equivalent to the given statement "If a line segment joins the center of a circle with a point on the circle, then it is a radius" is the *contrapositive* of the given statement.

What is the contrapostive of p -> q ?

Answer 3

student 3

Answer 4

I do not agree that student 3 is correct.

Answer 5

why I need help on this

Answer 6

That is why I asked this: What is the contrapostive of p -> q ?

Answer 7

a statement in the opposite order of an original statement with both parts negated.

Answer 8

ok, so what student said something that looks like that?

Answer 9

ok so student 2

Answer 10

i'm a little confused because sutdent number 1 is second in your list up there, what statement is the logical equivalent?

Answer 11

In your given statement, the following is true.
P: a line segment joins the center of a circle with a point on the circle
q: it is a radius

The above is p-> q.
The contrapositive of p->q is what you want for the logically equivalent statement.
The contrapositve, as you stated in words above, is written symbolically as:

~q ->~p {Read as not q implies not p}

Take the negation of: q: it is a radius and let it imply the negation of P: a line segment joins the center of a circle with a point on the circle.

That is the contrapositive and it is logically equivalent to the given implication. Logically equivalent means the two statements say the same thing.

Answer 12

So, which student do you think is correct?

Answer 13

@Napervillian I'll answer your questions after those of Carlos. Please do not add confusion. Thank you.

Answer 14

@Carlos2445 Student 2 is not correct.

You could get these correct every time if you would go with the contrapositive.

Answer 15

What you want is IF NOT (q: it is a radius ), then NOT ( P: a line segment joins the center of a circle with a point on the circle)
@Carlos2445

Answer 16

okay i got student 1
cause he mentions not twice

Answer 17

Student 1, I agree. That mentioning "not" twice can be tricky.

If the statement had been: If x = 4, then x^2 is not 9, then what would be the contrapositive?

Answer 18

if not x=4, then x^2 does not equal 9

Answer 19

No.
The contrapositive would be: If x^2 does NOT not equal 9, then x does NOT = 4.

Simplified: If x^2 =9, the x is not 4.

Answer 20

x=3