PQRS is a quadrilateral, PQ=7cm, QR=8cm and PR=7.5cm. (a) Calculate angle PSR (b) Hence if SR=SP, calculate angle SPR
(a) Well, if we have a quadrilateral, we know that the sum of the interior angles is equal to 360 degrees. Let's call angle PSR as x.
We know that angle PQR is a right angle (90 degrees), so the other two angles, PRQ and RPQ, should add up to 90 degrees.
PRQ + RPQ = 90
Since PQ = QR, we can say that angle PRQ is equal to angle RPQ.
Let's assume angle PRQ is y. So we have:
y + y = 90
2y = 90
y = 45 degrees
Now, we can find angle PSR. Since the sum of all angles in a quadrilateral is 360 degrees, we can say:
x + 45 + 90 + 90 = 360
x + 225 = 360
x = 360 - 225
x = 135 degrees
Therefore, angle PSR is 135 degrees.
(b) If SR = SP, it means that triangle SPR is an isosceles triangle.
Since we have angle SPR as x (135 degrees) and SR = SP, the other two angles, SRR and SPP, should be equal.
Let's call both these angles as z. So we have:
z + z + x = 180
2z + 135 = 180
2z = 180 - 135
2z = 45
z = 45 / 2
z = 22.5 degrees
Therefore, angle SPR is equal to 22.5 degrees.
(a) To calculate angle PSR, we can use the Law of Cosines. The formula is:
c^2 = a^2 + b^2 - 2ab*cos(C)
In the given quadrilateral PQRS:
PQ = 7 cm,
QR = 8 cm,
PR = 7.5 cm.
Using the Law of Cosines, we can calculate angle PSR as follows:
PR^2 = PQ^2 + QR^2 - 2*PQ*QR*cos(PSR)
Substituting the given values, we have:
7.5^2 = 7^2 + 8^2 - 2*7*8*cos(PSR)
56.25 = 49 + 64 - 112*cos(PSR)
56.25 = 113 - 112*cos(PSR)
112*cos(PSR) = 113 - 56.25
112*cos(PSR) = 56.75
cos(PSR) = 56.75/112
cos(PSR) ≈ 0.50759
To find angle PSR, we take the inverse cosine (cos^-1) of 0.50759:
PSR ≈ cos^-1(0.50759)
PSR ≈ 59.07 degrees
Therefore, angle PSR is approximately 59.07 degrees.
(b) If SR = SP, then we have an isosceles triangle SPR. In an isosceles triangle, the base angles are equal.
Since angle PSR is 59.07 degrees, the base angles SPR and SRP are also equal.
Therefore, angle SPR ≈ angle SRP ≈ 1/2(180 - 59.07) degrees.
angle SPR ≈ angle SRP ≈ 1/2(120.93) degrees
angle SPR ≈ angle SRP ≈ 60.47 degrees.
Hence, angle SPR is approximately 60.47 degrees.
To solve this problem, we will use the cosine rule and properties of quadrilaterals.
(a) To calculate angle PSR, we need to use the cosine rule. The cosine rule states that for a triangle with sides a, b, and c, and angle A opposite to side a, the following formula applies:
c^2 = a^2 + b^2 - 2ab*cos(A)
In our case, we have a quadrilateral PQRS with sides PQ = 7cm, QR = 8cm, and PR = 7.5cm. We are interested in finding angle PSR. To use the cosine rule, we need to find the side SR.
Using the fact that a quadrilateral is a closed figure, we know that the sum of the opposite sides is equal:
PQ + SR = QR + PR
Substituting the given values:
7cm + SR = 8cm + 7.5cm
SR = 8cm + 7.5cm - 7cm = 15.5cm
Now we can apply the cosine rule to triangle PSR (remember that both PS and SR now have a length of 15.5cm):
PR^2 = PS^2 + SR^2 - 2PS*SR*cos(PSR)
Solving for PSR:
7.5^2 = 15.5^2 + 15.5^2 - 2*15.5*15.5*cos(PSR)
56.25 = 240.25 + 240.25 - 480.25*cos(PSR)
56.25 = 480.5 - 480.25*cos(PSR)
480.25*cos(PSR) = 480.5 - 56.25
480.25*cos(PSR) = 424.25
cos(PSR) = 424.25 / 480.25
cos(PSR) ≈ 0.88 (rounded to two decimal places)
To find angle PSR, we can take the inverse cosine (cos^-1) of 0.88:
PSR ≈ cos^-1(0.88)
Using a calculator, we find that PSR ≈ 27.4 degrees (rounded to one decimal place).
(b) Given that SR = SP, we can deduce that triangle SPR is an isosceles triangle. In an isosceles triangle, the base angles are equal. Therefore, angle SPR is equal to angle SRP.
Since we already calculated angle PSR in part (a) to be approximately 27.4 degrees, we can conclude that angle SPR is also approximately 27.4 degrees.
In triangle PQR, use the law of cosines to determine the angles.
Unfortunately, just knowing that triangle PSR is isocseles tells you nothing about its other sides or angles.