The lengths, in inches, of the diagonals of five rectangles are given. Which lengths are irrational numbers?

Question

Select two correct answers.

A
150\sqrt{150}
150
​


B
144\sqrt{144}
144
​


C
121\sqrt{121}
121
​


D
100\sqrt{100}
100
​


E
111\sqrt{111}
111
​

Answer 1

To determine which lengths are irrational numbers, we can analyze each option.

1. Option A: \(150\sqrt{150}\):

   • \(150\) is a rational number.
   • \(\sqrt{150}\) is irrational (since \(150\) is not a perfect square).
   • The product \(150\sqrt{150}\) is irrational.

2. Option B: \(144\sqrt{144}\):

   • \(144\) is a rational number.
   • \(\sqrt{144} = 12\), which is rational.
   • The product \(144 \cdot 12\) is rational.

3. Option C: \(121\sqrt{121}\):

   • \(121\) is a rational number.
   • \(\sqrt{121} = 11\), which is rational.
   • The product \(121 \cdot 11\) is rational.

4. Option D: \(100\sqrt{100}\):

   • \(100\) is a rational number.
   • \(\sqrt{100} = 10\), which is rational.
   • The product \(100 \cdot 10\) is rational.

5. Option E: \(111\sqrt{111}\):

   • \(111\) is a rational number.
   • \(\sqrt{111}\) is irrational (since \(111\) is not a perfect square).
   • The product \(111\sqrt{111}\) is irrational.

Based on this analysis, the options that correspond to irrational lengths are:

• Option A: \(150\sqrt{150}\)
• Option E: \(111\sqrt{111}\)

So the correct answers are A and E.