Question

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



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**.