The correct equation to determine the length of QR using the law of cosines in triangle QRS is QR^2 = 20^2 + 27^2 - 2(20)(27) * cos(99°). Here option A is correct.
The length of QR using the law of cosines:
Identify the triangle and given information:
Triangle: QRS
Sides: PS = 20, QS = 27, angle R = 99°
We need to find the side QR.
Apply the law of cosines:
The law of cosines states: QR^2 = PS^2 + QS^2 - 2 * PS * QS * cos(R)
Substitute the given values:
QR^2 = 20^2 + 27^2 - 2 * 20 * 27 * cos(99°)
Calculate the right side:
QR^2 = 400 + 729 - 1080 * cos(99°) (approximately)
Solve for QR:
Take the square root of both sides (remember that the square root of a negative number is not a real number, so ensure the result is positive):
QR = √(400 + 729 - 1080 * cos(99°)) (approximately)
Calculate the numerical value:
Using a calculator or software, you'll get QR ≈ 10.77 units (approximately).
Therefore, the length of QR in triangle QRS is approximately 10.77 units. This confirms that option A (QR^2 = 20^2 + 27^2 - 2(20)(27) * cos(99°)) is the correct equation to use and provides the accurate value for QR. Here option A is correct.