IXL - Counterexamples (Geometry practice)

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

If a tool is used for measuring, then it is a ruler.

a ruler

a tape measure

a hammer

a pair of scissors

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question

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

If a number is a perfect square, then the number has exactly three factors.

100

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 number is a perfect square,	then	the number has exactly three factors.
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 10. The hypothesis is false, because 10 is not a perfect square. A counterexample must make the hypothesis true, so this is not a counterexample.

Next try 30. The hypothesis is false, because 30 is not a perfect square. A counterexample must make the hypothesis true, so this is not a counterexample.

Next try 100. The hypothesis is true, because 100 is a perfect square. The conclusion is false, because 100 has more than three factors. A counterexample makes the hypothesis true and the conclusion false. So, this is a counterexample.

Finally try 99. The hypothesis is false, because 99 is not a perfect square. A counterexample must make the hypothesis true, so this is not a counterexample.

In summary, the counterexample is 100.

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A.1Identify hypotheses and conclusions

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