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Find the radius of circle π given π΄πΆ is equal to 14 centimeters and angle π΄ is equal to 50 degrees.
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Give the answer to two decimal places.
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The line π΄π΅ is a diameter of the circle.
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Therefore, the radius will be half of this length.
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We also know that the angle πΆ is equal to 90 degrees.
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This is because any angle in a semicircle is equal to 90 degrees.
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If we now consider the triangle π΄π΅πΆ, we can use right-angle trigonometry or SOHCAHTOA to calculate the length of π΄π΅.
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The length π΅πΆ is opposite the 50-degree angle.
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The length π΄πΆ, which weβre told is 14 centimeters, is adjacent to the 50-degree angle.
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Finally, the length π΄π΅ is the hypotenuse as it is the longest side of the right-angle triangle.
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We know the length of the adjacent.
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And weβre trying to calculate the hypotenuse.
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Therefore, we will use the cosine ratio.
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This states that the cos of angle π is equal to the adjacent divided by the hypotenuse.
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Substituting in our values from the diagram gives us cos 50 is equal to 14 divided by the length π΄π΅.
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Multiplying both sides of this equation by π΄π΅ and dividing by cos 50 tells us that the length π΄π΅ is equal to 14 divided by cos 50.
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Typing this into the calculator gives us a value for π΄π΅ equal to 21.7801 and so on.
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We can therefore say that the diameter of circle π is 21.78 centimeters to two decimal places.
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As the radius is half of the diameter, we can divide this number by two.
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21.78 divided by two is equal to 10.89.
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This means that the radius of circle π is 10.89 centimeters to two decimal places.