Question

select all the correct answers.
which three pairs of side lengths are possible measurements for the triangle?
image of a right triangle with a right angle at b, angle at a is 45°, angle at c is 45°

• bc = 10√3, ac = 20
• bc = 10, ac = 10√2
• ab = 9√2, ac = 18
• ab = 14, bc = 14
• ab = 7, bc = 7√3
• ab = 9, ac = 18

Explanation:

Step 1: Identify Triangle Type

The triangle is a right - isosceles triangle (since two angles are \(45^{\circ}\) and one is \(90^{\circ}\)). In a right - isosceles triangle, the legs (\(AB\) and \(BC\)) are equal, and the hypotenuse (\(AC\)) is related to the legs by the formula \(AC=\text{leg}\times\sqrt{2}\) or \(\text{leg}=\frac{AC}{\sqrt{2}}=\frac{AC\sqrt{2}}{2}\)

Step 2: Analyze Each Option

- **Option 1: \(BC = 10\sqrt{3}, AC = 20\)**
If \(BC\) is a leg and \(AC\) is the hypotenuse, then the leg should be \(\frac{AC}{\sqrt{2}}=\frac{20}{\sqrt{2}} = 10\sqrt{2} eq10\sqrt{3}\). So this option is incorrect.
- **Option 2: \(BC = 10, AC = 10\sqrt{2}\)**
Since \(BC\) is a leg and \(AC\) is the hypotenuse, and \(\text{leg}\times\sqrt{2}=10\times\sqrt{2}=AC\), this option is correct. Also, since it is an isosceles right - triangle, \(AB = BC = 10\) which is consistent with the triangle's properties.
- **Option 3: \(AB = 9\sqrt{2}, AC = 18\)**
If \(AB\) is a leg, then the hypotenuse \(AC\) should be \(AB\times\sqrt{2}=9\sqrt{2}\times\sqrt{2}=18\), which matches the given \(AC = 18\). So this option is correct.
- **Option 4: \(AB = 14, BC = 14\)**
In a right - isosceles triangle, equal legs (\(AB\) and \(BC\)) are a characteristic property. So this option is correct as \(AB = BC\) (both legs)
- **Option 5: \(AB = 7, BC = 7\sqrt{3}\)**
In a right - isosceles triangle, the two legs should be equal. But \(AB = 7\) and \(BC = 7\sqrt{3}\) are not equal. So this option is incorrect.
- **Option 6: \(AB = 9, AC = 18\)**
If \(AB\) is a leg, then the hypotenuse \(AC\) should be \(AB\times\sqrt{2}=9\sqrt{2}\approx12.73 eq18\)? Wait, no. Wait, we made a mistake earlier. Wait, in a \(45 - 45-90\) triangle, \(AC=\text{leg}\times\sqrt{2}\). If \(AB = 9\) (leg), then \(AC = 9\sqrt{2}\approx12.73\)? Wait, no, wait, let's re - calculate. Wait, if \(AB\) is the leg and \(AC\) is the hypotenuse, then \(AC = AB\sqrt{2}\). But if \(AB = 9\), \(AC=9\sqrt{2}\approx12.73 eq18\). Wait, we made an error in the initial analysis of this option. Wait, no, wait, maybe we mixed up the formula. Wait, in a right - isosceles triangle, \(\sin(45^{\circ})=\frac{\text{leg}}{\text{hypotenuse}}\), so \(\text{hypotenuse}=\frac{\text{leg}}{\sin(45^{\circ})}=\text{leg}\times\sqrt{2}\), and \(\cos(45^{\circ})=\frac{\text{leg}}{\text{hypotenuse}}\) (same as \(\sin(45^{\circ})\) for \(45^{\circ}\) angle). Wait, let's re - check the option \(AB = 9, AC = 18\). If \(AB\) is the leg, then \(\sin(45^{\circ})=\frac{AB}{AC}\), so \(AC=\frac{AB}{\sin(45^{\circ})}=\frac{9}{\frac{\sqrt{2}}{2}}=9\sqrt{2}\approx12.73 eq18\). Wait, we had a mistake in the initial analysis. Let's re - evaluate all options correctly.

Step 2 (Re - evaluation):

- **Option 1: \(BC = 10\sqrt{3}, AC = 20\)**
\(\sin(45^{\circ})=\frac{BC}{AC}\), \(\frac{\sqrt{2}}{2}=\frac{10\sqrt{3}}{20}=\frac{\sqrt{3}}{2}\), which is false.
- **Option 2: \(BC = 10, AC = 10\sqrt{2}\)**
\(\sin(45^{\circ})=\frac{BC}{AC}=\frac{10}{10\sqrt{2}}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}\), which is true.
- **Option 3: \(AB = 9\sqrt{2}, AC = 18\)**
\(\sin(45^{\circ})=\frac{AB}{AC}=\frac{9\sqrt{2}}{18}=\frac{\sqrt{2}}{2}\), which is true.
- **Option 4: \(AB = 14, BC = 14\)**
Since \(AB = BC\) (legs of right - isosceles triangle) and \(\angle B = 90^{\circ}\), this is consistent with the triangle's properties. Also, \(\sin(45^{\circ})=\frac{AB}{AC}\), if \(AB = 14\), \(AC = 14\sqrt{2}\), but the option only checks the equality of the legs, which is a necessary condition for a right - isosceles triangle. So this option is correct.
- **Option 5: \(AB = 7, BC = 7\sqrt{3}\)**
\(AB eq BC\), so it can't be a right - isosceles triangle. Incorrect.
- **Option 6: \(AB = 9, AC = 18\)**
\(\sin(45^{\circ})=\frac{AB}{AC}=\frac{9}{18}=\frac{1}{2} eq\frac{\sqrt{2}}{2}\). Incorrect. Wait, we had a mistake in the first re - evaluation. The correct formula for the right - isosceles triangle: legs \(a = b\), hypotenuse \(c=a\sqrt{2}\)

Let's use the correct formula \(c = a\sqrt{2}\) (where \(a\) is the length of the leg, \(c\) is the hypotenuse)
- For \(BC = 10, AC = 10\sqrt{2}\): Let \(a = BC = 10\), then \(c=a\sqrt{2}=10\sqrt{2}=AC\), correct.
- For \(AB = 14, BC = 14\): Here \(a = AB = BC = 14\), so it is a right - isosceles triangle, correct.
- For \(AB = 9, AC = 18\): Let \(a = AB = 9\), then \(c=a\sqrt{2}=9\sqrt{2}\approx12.73 eq18\). Wait, no, if we consider the angle at \(A\) is \(45^{\circ}\), \(\cos(45^{\circ})=\frac{AB}{AC}\), \(\frac{\sqrt{2}}{2}=\frac{9}{18}=\frac{1}{2}\), no. Wait, the angle at \(A\) is \(45^{\circ}\), so \(\cos(45^{\circ})=\frac{AB}{AC}\), \(\sin(45^{\circ})=\frac{BC}{AC}\). Since it is an isosceles right - triangle, \(AB = BC\), so \(\cos(45^{\circ})=\sin(45^{\circ})=\frac{AB}{AC}\)

The correct options are:
- \(BC = 10, AC = 10\sqrt{2}\) (because \(10\sqrt{2}=10\times\sqrt{2}\), leg \(BC = 10\), hypotenuse \(AC = 10\sqrt{2}\))
- \(AB = 14, BC = 14\) (legs are equal, so it is a right - isosceles triangle)
- \(AB = 9\sqrt{2}, AC = 18\) (since \(\frac{AB}{AC}=\frac{9\sqrt{2}}{18}=\frac{\sqrt{2}}{2}=\cos(45^{\circ})\), which is consistent with the angle at \(A\) being \(45^{\circ}\))

Answer:

• \( BC = 10, AC = 10\sqrt{2} \)
• \( AB = 14, BC = 14 \)
• \( AB = 9, AC = 18 \)