PQRS in which PQ=PS=5cm RS=5.5cm angle p=90 degree and angle S =120 degree
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In the given figure, PQRS is a quadrilateral where PQ = PS = 5 cm, RS = 5.5 cm, angle P = 90 degrees, and angle S = 120 degrees.
To find the length QR, we can use the Pythagorean theorem and trigonometric ratios.
Step 1: Use the Pythagorean theorem to find the length of the diagonal PR.
In right triangle PQR, we have PQ = 5 cm and angle P = 90 degrees.
Using the Pythagorean theorem, we can find QR as follows:
QR^2 = PR^2 - PQ^2
QR^2 = RS^2 - PS^2
QR^2 = (5.5 cm)^2 - (5 cm)^2
QR^2 = 30.25 cm^2 - 25 cm^2
QR^2 = 5.25 cm^2
QR = sqrt(5.25 cm^2)
QR = 2.29 cm
Step 2: Use the law of cosines to find angle Q, which is opposite the side QR.
The law of cosines states: c^2 = a^2 + b^2 - 2ab*cos(C), where c is the side opposite the angle C.
In triangle PQS, we have PQ = 5 cm, PS = 5 cm, and angle S = 120 degrees.
Using the law of cosines, we can find angle Q as follows:
QR^2 = PQ^2 + PS^2 - 2*PQ*PS*cos(Q)
(2.29 cm)^2 = (5 cm)^2 + (5 cm)^2 - 2*(5 cm)*(5 cm)*cos(Q)
5.2441 cm^2 = 50 cm^2 - 50 cm^2*cos(Q)
cos(Q) = (50 cm^2 - 5.2441 cm^2)/(50 cm^2)
cos(Q) = 0.894882
Q = cos^(-1)(0.894882)
Q ≈ 27.33 degrees
Step 3: Find the length QR using the law of sines.
The law of sines states: a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are the sides opposite the angles A, B, C respectively.
In triangle PQS, we have PQ = 5 cm, PS = 5 cm, Q = 27.33 degrees, and angle S = 120 degrees.
Using the law of sines, we can find QR as follows:
QR/sin(Q) = PS/sin(S)
QR/sin(27.33 degrees) = 5 cm/sin(120 degrees)
QR/sin(27.33 degrees) = 5 cm/sin(120 degrees)
QR ≈ 2.63 cm
Therefore, the length QR is approximately 2.63 cm.