- Verify the triangle is a right triangle using the Pythagorean theorem, confirming \(7^2+24^2=25^2\).
- Calculate the angles \(A\) and \(B\) using the inverse sine trigonometric function.
- Find \(m\angle A = \arcsin(\frac{24}{25}) \approx 73.7^{\circ}\) and \(m\angle B = \arcsin(\frac{7}{25}) \approx 16.3^{\circ}\).
- Conclude that the correct answer matches the values given in option 2.
- The final answer is \(\boxed{2}\).
Explanation
- Analyze the triangle and identify the right angle.
In a right triangle \(ABC\) with side lengths \(AC=7\), \(BC=24\), and \(AB=25\), we first confirm the right angle property by checking the Pythagorean theorem: \(AC^2 + BC^2 = 7^2 + 24^2 = 49 + 576 = 625\). Since \(AB^2 = 25^2 = 625\), the triangle is indeed a right triangle with the right angle at \(C\), so \(m\angle C = 90^{\circ}\).
- Calculate the angles using trigonometric ratios.
To find the angles \(A\) and \(B\), we use trigonometric ratios for a right triangle:
- For angle \(A\), the opposite side is \(BC=24\) and the hypotenuse is \(AB=25\). Thus, \(\sin(A) = \frac{BC}{AB} = \frac{24}{25} = 0.96\).
- Using the inverse sine function, \(m\angle A = \arcsin(0.96) \approx 73.739795^{\circ}\), which rounds to \(73.7^{\circ}\).
- For angle \(B\), the opposite side is \(AC=7\) and the hypotenuse is \(AB=25\). Thus, \(\sin(B) = \frac{AC}{AB} = \frac{7}{25} = 0.28\).
- Using the inverse sine function, \(m\angle B = \arcsin(0.28) \approx 16.260205^{\circ}\), which rounds to \(16.3^{\circ}\).
- Identify the correct option.
Comparing our results (\(m\angle A \approx 73.7^{\circ}\), \(m\angle B \approx 16.3^{\circ}\), \(m\angle C = 90^{\circ}\)) with the given options, we find that these values correspond exactly to option 2. Thus, option 2 is the correct choice.
Examples
Understanding how to calculate angles in a right triangle is essential in fields like construction and navigation. For example, a carpenter building a custom roof uses the ratios of the triangle's sides to determine the correct angle for the rafters, ensuring the structure is sound and aesthetically pleasing. By using trigonometric functions like \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\), one can precisely calculate any angle given the lengths of the sides. This foundation in geometry allows for accurate measurements in engineering, architecture, and even everyday DIY home projects. For our triangle, we found the angles by calculating \(m\angle A = \arcsin\left(\frac{24}{25}\right) \approx 73.7^{\circ}\) and \(m\angle B = \arcsin\left(\frac{7}{25}\right) \approx 16.3^{\circ}\).