In rectangle PQRS, diagonal PR makes a 60 degrees angle with side PS. If PR =10, what is the area of the rectangle?
Hint: Think of the ubiquitous 3-4-5 right triangle, scaled up to 6-8-10.
Nope. Forgot the 60 degree angle. In that case, recall that cos 60 = 1/2
To find the area of the rectangle PQRS, we need to know the length and width of the rectangle. From the given information, we know that diagonal PR makes a 60-degree angle with side PS, and PR = 10.
To solve this, we can use trigonometry and the relationship between the sides and angles of a right triangle.
Let's label the length of the rectangle as PQ and the width as PS. Therefore, the diagonal PR will act as the hypotenuse of the right triangle PQR.
Using trigonometry, we can determine the relationship between the sides of the triangle and the given angle:
In a right triangle, the side opposite the angle is called the opposite side (PS in this case), and the side adjacent to the angle is called the adjacent side (PQ in this case).
We have the opposite side (PS) and the hypotenuse (PR), so we can use the sine function:
sin(60 degrees) = Opposite / Hypotenuse
sin(60 degrees) = PS / PR
sin(60 degrees) = PS / 10
Using the sine of 60 degrees (which is √3/2), we can solve for PS:
√3/2 = PS / 10
Cross-multiplying to solve for PS:
2PS = 10√3
PS = 5√3
Now that we know the width (PS) and the length (PR) of the rectangle, we can find the area:
Area of the rectangle = Length × Width
Area of the rectangle = PR × PS
Area of the rectangle = 10 × 5√3
Area of the rectangle = 50√3
Therefore, the area of the rectangle PQRS is 50√3.