Question

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

If a line intersects a circle at exactly two points, then it is a secant.

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

Student 4: If a line is not a secant, then it does not intersect the circle at exactly two points.

Student 3: If a line is not a secant, then it intersects the circle at exactly two points.

Student 1: If a line intersects a circle at exactly two points, then it is not a secant

Answer 1

Student 2: If a line does not intersect a circle at exactly two points, then it is a secant.

Answer 2

the correct answer would be a right @mathmate

Answer 3

If (a) represents student 4's answer, yes, because it is the contrapositive of the given statement.
If we invert the condition and consequence, and put them both in negation of the original, then it is the contrapositive and is logically equivalent.

Answer 4

Read the following statements:

Statement 1: "If she is stuck in traffic, then she is late."
Statement 2: "If she is late, then she is stuck in traffic."
Statement 3: "If she is not late, then she is not stuck in traffic."

Ruby writes, "Statement 3 is the inverse of statement 2 and contrapositive of statement 1."
Christine writes, "Statement 1 is the inverse of statement 2 and converse of statement 3."

Which option is true?

Both Ruby and Christine are correct.

Both Ruby and Christine are incorrect.

Only Ruby is correct.

Only Christine is correct.

Answer 5

ok so for this one it would be the same concept?

Answer 6

@mathmate

Answer 7

I'll read through the question and be right back.

Answer 8

Recall what @FriedRice wrote:
inverse: negation in both condition and consequence, or
\(\lnot p->\lnot q\) is the inverse of \(p->q\)
and
\(\lnot q->\lnot p\) is the contrapositive of \(p->q\)
So give it a try and see who (or both) of Ruby or Christine is right.

Answer 9

ok thanks

Answer 10

Read the statements shown below:

Statement 1: If it has two sides, then it is a polygon.
Statement 2: If it is not a polygon, then it does not have two sides.

Are the two statements logically equivalent?

No, both statements are false.

Yes, both statements are true.

No, only one statement is true.

Yes, both statements are false.

Answer 11

The 4 choices do not answer the question "if they are logically equivalent".
If you check if one is the contrapositive of the other you can decide if they are logically equivalent. That would answer the question.

As for the 4 choices, you have to decide if a polygon has two sides (but did not say exactly two sides, or at least two sides). Commonly in English, this may be interpreted as exactly two sides. Can you attempt this part?