Area of Isosceles Triangle Given Radius & Altitude

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In summary, the problem involves a circle with a radius of 5 and an isosceles triangle that is circumscribed within it. The altitude to the base of the triangle is 9, and the question is asking for the area of the triangle. However, the method of using the law of sines to find the two equal sides of the triangle cannot be applied without knowing one of the angles. The radii of the circle and the triangle's height and vertices must also be taken into consideration.
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studof09
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A circle has a radius of 5 and the circle is circumscribed around an isosceles triangle. If the length of the altitude to the base of the triangle is 9, what is the area of the triangle?

I tried to figure out the two equal sides of the triangle by using the law of sines but you cannot assume that you have two 45 degree angles and a 90 degree angle in the two triangles split apart by the altitude. I feel like I need to know one of the angles to solve this problem.
 
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Consider the radii of the circle and how they interact with the height of the triangle and the vertices.
 

1. What is the formula for finding the area of an isosceles triangle given the radius and altitude?

The formula for finding the area of an isosceles triangle given the radius and altitude is:
Area = (1/2) x (radius) x (altitude)

2. Can the area of an isosceles triangle be calculated if the base length is unknown?

Yes, the area of an isosceles triangle can still be calculated if the base length is unknown, as long as the radius and altitude are known. The formula for finding the area only requires the radius and altitude, not the base length.

3. How do you find the altitude of an isosceles triangle if the area and radius are known?

To find the altitude of an isosceles triangle, you can rearrange the formula for finding the area to solve for the altitude.
Altitude = (2 x area) / (radius)

4. Is the area of an isosceles triangle always the same regardless of the position of the triangle?

Yes, the area of an isosceles triangle is always the same regardless of the position of the triangle. This is because the area only depends on the length of the radius and the altitude, which remain constant regardless of the position of the triangle.

5. Can the area of an isosceles triangle be negative?

No, the area of an isosceles triangle cannot be negative. Area is a measure of the space occupied by a shape, and it cannot be negative. If the formula for finding the area of an isosceles triangle results in a negative value, it is likely a sign of an error in the calculation.

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