Question

question 2 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
bc = 7√2, ac = 14
ab = 6, ac = 6√2
bc = 8, ac = 8√3
ab = 15, bc = 15

Explanation:

Step1: Identify the triangle type

This is a right - angled isosceles triangle (\(45^{\circ}-45^{\circ}-90^{\circ}\) triangle). In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, the legs are equal (\(AB = BC\)), and the hypotenuse \(AC=\sqrt{2}\times\) leg length.

Step2: Check each option

• Option \(AB = 16,AC = 16\):

Since \(AC=\sqrt{2}AB\) in a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, if \(AB = 16\), then \(AC = 16\sqrt{2}
eq16\). This option is incorrect.

• Option \(AB = 11,AC = 22\):

If \(AB = 11\), then \(AC=\sqrt{2}\times11\approx15.56
eq22\). This option is incorrect.

• Option \(BC = 7\sqrt{2},AC = 14\):

Let \(AB = BC = 7\sqrt{2}\). Then \(AC=\sqrt{(7\sqrt{2})^{2}+(7\sqrt{2})^{2}}=\sqrt{98 + 98}=\sqrt{196}=14\). This option is correct.

• Option \(AB = 6,AC = 6\sqrt{2}\):

Let \(AB = 6\). Then \(AC=\sqrt{6^{2}+6^{2}}=\sqrt{36 + 36}=\sqrt{72}=6\sqrt{2}\). This option is correct.

• Option \(BC = 8,AC = 8\sqrt{3}\):

Let \(AB = BC = 8\). Then \(AC=\sqrt{8^{2}+8^{2}}=\sqrt{64 + 64}=\sqrt{128}=8\sqrt{2}
eq8\sqrt{3}\). This option is incorrect.

• Option \(AB = 15,BC = 15\):

Since \(AB = BC\) (legs of a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle), this satisfies the property of a right - angled isosceles triangle. This option is correct.

Answer:

C. \(BC = 7\sqrt{2},AC = 14\)；E. \(AB = 6,AC = 6\sqrt{2}\)；F. \(AB = 15,BC = 15\)