A triangle having an area of 216 squared inches is inscribed in a circle of radius 15 inches. If one of the sides of the triangle is 30 inches, find one of the other two sides.
Understand the Problem
The question is asking to find one of the other two sides of a triangle given its area, the radius of the circumscribed circle, and the length of one side. We can use the triangle area formula and properties of the triangle inscribed in a circle to solve for the unknown side.
Answer
The formula for one unknown side \( b \) is \( b = \frac{4AR}{a} \).
Answer for screen readers
The formula derived for one unknown side ( b ) is: $$ b = \frac{4AR}{a} $$
Steps to Solve
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Identify the given information The problem gives the area of the triangle ( A ), the radius of the circumscribed circle ( R ), and the length of one side ( a ).
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Use the formula for the area of a triangle The area of a triangle can also be expressed using the circumradius ( R ) and one side ( a ): $$ A = \frac{abc}{4R} $$ Where ( b ) and ( c ) are the other two sides of the triangle. Since we are trying to find one of the unknown sides, we'll rearrange this formula.
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Rearrange the area formula We want to isolate one side of the triangle. Rearranging gives: $$ bc = \frac{4AR}{a} $$
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Express one known side in terms of the others Since we need to find one of the unknown sides, we can express ( c ) in terms of ( b ): $$ c = \frac{4AR}{ab} $$
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Solve for ( b ) or ( c ) as needed If you know one of the two sides, substitute that value in and solve. If both sides are unknown, additional information would be needed to progress further.
The formula derived for one unknown side ( b ) is: $$ b = \frac{4AR}{a} $$
More Information
In a triangle, there are relationships like this that connect the area, sides, and the radius of the circumscribed circle, making it a fascinating aspect of geometry to explore. It illustrates how different properties of triangles relate to one another.
Tips
A common mistake is to forget to use consistent units for area and side lengths. Ensure that the area ( A ) should be in the same unit system as the side lengths ( a, b, c ) and the circumradius ( R ). Another mistake is not rearranging properly: always check your algebra when isolating variables.
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