In triangle \( ABC \), where \( AB \) is the hypotenuse labeled as \( 10 \), \( BC \) is the height labeled as \( a \), and \( CA \) is the base labeled as \( b \), we know that the angle opposite side \( BC \) (height) is \( 30^\circ \).
Since triangle \( ABC \) is a right triangle, we can use the relationship of the sides in a right triangle along with trigonometric functions.
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Using the sine function to find \( a \): \[ \sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{10} \] The sine of \( 30^\circ \) is \( \frac{1}{2} \): \[ \frac{1}{2} = \frac{a}{10} \] To find \( a \), we can multiply both sides by \( 10 \): \[ a = 10 \cdot \frac{1}{2} = 5 \]
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Using the cosine function to find \( b \): \[ \cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{10} \] The cosine of \( 30^\circ \) is \( \frac{\sqrt{3}}{2} \): \[ \frac{\sqrt{3}}{2} = \frac{b}{10} \] To find \( b \), we multiply both sides by \( 10 \): \[ b = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3} \]
Thus, the exact values are: \[ a = 5, \quad b = 5\sqrt{3}. \]
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