In triangle ABC, where AB is the hypotenuse measuring 7 units, BC is the height (labeled as \( a \)), CA is the base (labeled as \( b \)), and the angle opposite side BC (angle A) is 30 degrees, we can use trigonometric ratios to find the values for \( a \) and \( b \).
To find \( a \):
Using the sine function, which relates the opposite side to the hypotenuse in a right triangle:
\[ \sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AB} \]
Substituting the known values:
\[ \sin(30^\circ) = \frac{a}{7} \]
Since \( \sin(30^\circ) = \frac{1}{2} \), we can write:
\[ \frac{1}{2} = \frac{a}{7} \]
Now, solving for \( a \):
\[ a = 7 \times \frac{1}{2} = 3.5 \]
To find \( b \):
Using the cosine function, which relates the adjacent side to the hypotenuse:
\[ \cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{CA}{AB} \]
Substituting the known values:
\[ \cos(30^\circ) = \frac{b}{7} \]
Knowing \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \), we can write:
\[ \frac{\sqrt{3}}{2} = \frac{b}{7} \]
Now, solving for \( b \):
\[ b = 7 \times \frac{\sqrt{3}}{2} = \frac{7\sqrt{3}}{2} \]
Final Values
Thus, the exact values are:
\[ a = 3.5 \]
\[ b = \frac{7\sqrt{3}}{2} \]
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