An isosceles triangle is to be drawn with its two equal sides of legth 10. What is the maximum possible area for the triangle? [Note: The area of a triangle is given by half the product of two sides with the sine of the included angle.]
To find the maximum possible area of the isosceles triangle, we need to determine the measure of the included angle between the two equal sides.
Let's denote the included angle as θ.
To maximize the area, we need to find the value of θ that maximizes the sine of the included angle. The sine function is maximized when the angle is 90 degrees.
In an isosceles triangle, the two base angles are equal. So, let's denote each base angle as α.
Since the sum of the angles in a triangle is 180 degrees, we can determine α as follows:
180 degrees = 2α + θ
Simplifying the equation:
2α = 180 degrees - θ
α = (180 degrees - θ)/2
The area of a triangle is given by half the product of the two equal sides with the sine of the included angle:
Area = (1/2) * (10) * (10) * sin(θ)
To find the maximum area, we substitute the value of θ that maximizes the sine function. Since sin(90 degrees) = 1, the maximum area occurs when θ is 90 degrees.
Therefore, the maximum possible area for the isosceles triangle is:
Area = (1/2) * (10) * (10) * sin(90 degrees)
= (1/2) * (10) * (10) * 1
= 50 square units