If the side opposite a 60-degree angle in a 30-60-90 right triangle is 73–√ cm

Question

7
3
cm
long, how long is the hypotenuse?(1 point)
Responses

7 cm
7 cm

14 cm
14 cm

73–√ cm
7
3
cm
7 Start Root 3 End Root cm

28 cm

Answer 1

In a 30-60-90 triangle, the sides are in a specific ratio:

• The side opposite the 30-degree angle (the shortest side) is \(x\).
• The side opposite the 60-degree angle is \(x\sqrt{3}\).
• The hypotenuse is \(2x\).

In your case, the side opposite the 60-degree angle is given as \(73 - \sqrt{cm}\).

Let’s denote the length of the side opposite the 30-degree angle as \(x\). Therefore, we can set up the following equation based on the side opposite the 60-degree angle:

\[ x\sqrt{3} = 73 - \sqrt{cm} \]

To find \(x\), we can rearrange this equation:

\[ x = \frac{73 - \sqrt{cm}}{\sqrt{3}} \]

The hypotenuse, which is \(2x\), can be found by multiplying \(x\) by 2:

\[ \text{Hypotenuse} = 2x = 2 \cdot \frac{73 - \sqrt{cm}}{\sqrt{3}} = \frac{2(73 - \sqrt{cm})}{\sqrt{3}} = \frac{146 - 2\sqrt{cm}}{\sqrt{3}} \]

However, it seems that the available response options do not match this expression.

If you're looking for the numerical value of the hypotenuse based solely on the length of the side opposite the 60-degree angle \(73 - \sqrt{cm}\), you can derive the hypotenuse by substituting the value of the side length into the 30-60-90 triangle ratios.

Assuming \(73 - \sqrt{cm}\) represents the side opposite the 60-degree angle:

1. Finding the length of the hypotenuse: \[ \text{Hypotenuse} = 2 \left(\frac{73 - \sqrt{cm}}{\sqrt{3}}\right) \] But you want to determine a numerical value based on the answers provided. Given that \(\sqrt{3} \approx 1.732\), one can assess the values.

To simply compute for valid answers:

• \(x\) opposite the 30° angle thus measures as \(x \approx \frac{73 - \sqrt{cm}}{\sqrt{3}}\)

After approximations, you must assess valid outcomes:

1. 7 cm - Improbable based on higher lengths

2. 14 cm - Also likely insufficient derived via factors

3. 73 - √cm cm - Close but requiring further clarifying expressions confusing use metrics of cm.

4. 28 cm - Seems plausible if all hypotenuse derivation fits correctly through.

Conclusively, 28 cm could perhaps be the best estimate if validations hold. Let’s affirm metrics through direct evaluations for best congruence based on ratios or dimensional measures guiding throughout significant triangle structures.

Thus, the most suitable answer is 28 cm.