Question

question 1 of 5

select all the correct answers.

which three pairs of side lengths are possible measurements for the triangle?

\\(ab = 16\\), \\(ac = 16\\)
\\(ab = 11\\), \\(ac = 22\\)
\\(ab = 6\\), \\(ac = 6\sqrt{2}\\)
\\(bc = 8\\), \\(ac = 8\sqrt{3}\\)
\\(ab = 15\\), \\(bc = 15\\)
\\(bc = 7\sqrt{2}\\), \\(ac = 14\\)

Explanation:

Identify triangle properties

The given triangle \(\triangle ABC\) is a right triangle with \(\angle B = 90^\circ\).
Since \(\angle A = 45^\circ\) and \(\angle C = 45^\circ\), it is a \(45^\circ-45^\circ-90^\circ\) special right triangle.
This means \(\triangle ABC\) is an isosceles right triangle.

Determine side length relationships

For any \(45^\circ-45^\circ-90^\circ\) triangle, the legs are equal in length, and the hypotenuse is \(\sqrt{2}\) times the length of a leg:

$$ AB = BC $$

$$ AC = AB\sqrt{2} = BC\sqrt{2} $$

Evaluate the given options

We test each option against these relationships:

1. \(AB = 16, AC = 16\)

• Here, \(AB = AC\), which is incorrect because the hypotenuse \(AC\) must be longer than the leg \(AB\).

1. \(AB = 11, AC = 22\)

• If \(AB = 11\), then \(AC = 11\sqrt{2} \approx 15.56

eq 22\). This is incorrect.

1. \(AB = 6, AC = 6\sqrt{2}\)

• Since \(AC = AB\sqrt{2}\), this matches the ratio \(6 : 6\sqrt{2}\). This is correct.

1. \(BC = 8, AC = 8\sqrt{3}\)

• If \(BC = 8\), then \(AC = 8\sqrt{2}

eq 8\sqrt{3}\). This is incorrect.

1. \(AB = 15, BC = 15\)

• Since \(AB\) and \(BC\) are the two legs, they must be equal. This is correct.

1. \(BC = 7\sqrt{2}, AC = 14\)

• If \(BC = 7\sqrt{2}\), then \(AC = BC\sqrt{2} = (7\sqrt{2})\sqrt{2} = 7 \times 2 = 14\). This is correct.

Answer:

• \(AB = 16, AC = 16\)
• \(AB = 11, AC = 22\)
• \(AB = 6, AC = 6\sqrt{2}\) (Correct answer)
• \(BC = 8, AC = 8\sqrt{3}\)
• \(AB = 15, BC = 15\) (Correct answer)
• \(BC = 7\sqrt{2}, AC = 14\) (Correct answer)