IXL - Counterexamples (Algebra 1 practice)

Which of the following is a counterexample for this conditional statement?

If a pet is furry, then it is a hamster.

a mouse

a lizard

a frog

a goldfish

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Learn with an example

question

Which of the following is a counterexample for this conditional statement?

If a rectangle has a perimeter of 14, then its area is 10.

a rectangle with sides measuring 10 and 1

a rectangle with sides measuring 10 and 3

a rectangle with sides measuring 4 and 3

a rectangle with sides measuring 5 and 2

key idea

A conditional statement can be expressed as If A, then B.

A is the hypothesis and B is the conclusion.

A counterexample is an example in which the hypothesis is true, but the conclusion is false. If you can find a counterexample to a conditional statement, then that conditional statement is false.

solution

First notice that the given statement is in if/then form:

If	a rectangle has a perimeter of 14,	then	its area is 10.
If	A,	then	B.

Now go through the answer choices one by one, looking for the counterexample that makes the hypothesis true and the conclusion false.

First try a rectangle with sides measuring 10 and 1. The hypothesis is false, because it does not have a perimeter of 14. A counterexample must make the hypothesis true, so this is not a counterexample.

Next try a rectangle with sides measuring 10 and 3. The hypothesis is false, because it does not have a perimeter of 14. A counterexample must make the hypothesis true, so this is not a counterexample.

Next try a rectangle with sides measuring 4 and 3. The hypothesis is true, because it has a perimeter of 14. The conclusion is false, because it does not have an area of 10. A counterexample makes the hypothesis true and the conclusion false. So, this is a counterexample.

Finally try a rectangle with sides measuring 5 and 2. The hypothesis is true, because it has a perimeter of 14. The conclusion is also true, because it has an area of 10. A counterexample must make the conclusion false, so this is not a counterexample.

In summary, the counterexample is a rectangle with sides measuring 4 and 3.

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