The area of a triangle can be calculated using the formula: Area = 0.5 × base × height. However, when two sides of a triangle are known, we can also use another formula that incorporates the sine of the angle between those two sides.
The formula for the area when two sides and the included angle are used is: Area = 0.5 × a × b × sin(C), where:
- a and b are the lengths of the sides of the triangle, and
- C is the angle between those sides.
In this case, we have:
- a = 7
- b = 10
The area will be maximized when the value of sin(C) is at its highest, which occurs when C is 90 degrees (or π/2 radians). At this angle, sin(C) equals 1.
So, substituting the values into the area formula:
Area = 0.5 × 7 × 10 × sin(90°)
= 0.5 × 7 × 10 × 1
= 0.5 × 70
= 35Thus, the greatest possible area of the triangle with sides of lengths 7 and 10 units, when the included angle is 90 degrees, is 35 square units.