Question

which three pairs of side lengths are possible measurements for the triangle? ab = 9, ac = 18 ab = 9√2, ac = 18 bc = 10√3, ac = 20 bc = 10, ac = 10√2 ab = 14, bc = 14 ab = 7, bc = 7√3

Explanation:

Step1: Identify triangle type

This is a 45 - 45 - 90 right - triangle. In a 45 - 45 - 90 triangle, the ratio of the side lengths is $a:a:a\sqrt{2}$, where the legs (the sides opposite the 45 - degree angles) have length $a$ and the hypotenuse (the side opposite the 90 - degree angle) has length $a\sqrt{2}$. That is, if the length of each leg is $x$, the length of the hypotenuse $y$ satisfies $y = x\sqrt{2}$ or $x=\frac{y}{\sqrt{2}}=\frac{y\sqrt{2}}{2}$.

Step2: Check each option

Option 1: $AB = 9, AC = 18$

If $AB$ is a leg and $AC$ is the hypotenuse, then for a 45 - 45 - 90 triangle, if $AB=a = 9$, the hypotenuse should be $a\sqrt{2}=9\sqrt{2}
eq18$. So this option is incorrect.

Option 2: $AB = 9\sqrt{2}, AC = 18$

If $AB$ is a leg, then the hypotenuse $AC$ should be $AB\sqrt{2}$. Since $AB = 9\sqrt{2}$, then $AB\sqrt{2}=9\sqrt{2}\times\sqrt{2}=18$. This option is correct.

Option 3: $BC = 10\sqrt{3}, AC = 20$

If $BC$ is a leg and $AC$ is the hypotenuse, then if $BC=a = 10\sqrt{3}$, the hypotenuse should be $a\sqrt{2}=10\sqrt{6}
eq20$. So this option is incorrect.

Option 4: $BC = 10, AC = 10\sqrt{2}$

If $BC$ is a leg and $AC$ is the hypotenuse, and for a 45 - 45 - 90 triangle, if $BC=a = 10$, then the hypotenuse $AC=a\sqrt{2}=10\sqrt{2}$. This option is correct.

Option 5: $AB = 14, BC = 14$

Since the two legs of a 45 - 45 - 90 triangle can be equal, this option is correct.

Option 6: $AB = 7, BC = 7\sqrt{3}$

In a 45 - 45 - 90 triangle, the ratio of the legs is $1:1$, not $1:\sqrt{3}$. So this option is incorrect.

Answer:

$AB = 9\sqrt{2}, AC = 18$
$BC = 10, AC = 10\sqrt{2}$
$AB = 14, BC = 14$