The length of the side QR of a triangle PQR=14.5cm PqR 71° QPR =57. Find the length of side PR and PQ
so now we know that angle 52°
PR/sin71° = 14.5/sin57° = PQ/sin52°
Now just plug and chug
To find the length of side PR and PQ in the triangle PQR, we can use the Law of Sines.
The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
Let's use this law to find the length of side PR first, using side QR and its opposite angle QPR.
Step 1: Start with the formula of the Law of Sines:
PR / sin(QPR) = QR / sin(PQR)
Step 2: Substitute the known values into the equation:
PR / sin(57°) = 14.5 cm / sin(71°)
Step 3: Solve for PR.
PR = (sin(57°) / sin(71°)) * 14.5 cm
Using a calculator, we get:
PR ≈ (0.8293 / 0.9397) * 14.5 cm
PR ≈ 0.8833 * 14.5 cm
PR ≈ 12.82 cm
So, the length of side PR is approximately 12.82 cm.
Now, let's find the length of side PQ using side QR and its opposite angle PQR.
Step 1: Start with the formula of the Law of Sines:
PQ / sin(PQR) = QR / sin(QPR)
Step 2: Substitute the known values into the equation:
PQ / sin(71°) = 14.5 cm / sin(57°)
Step 3: Solve for PQ.
PQ = (sin(71°) / sin(57°)) * 14.5 cm
Using a calculator, we get:
PQ ≈ (0.9397 / 0.8293) * 14.5 cm
PQ ≈ 1.1339 * 14.5 cm
PQ ≈ 16.42 cm
So, the length of side PQ is approximately 16.42 cm.
To find the length of side PR in triangle PQR, we can use the Law of Cosines. The formula for the Law of Cosines is:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where:
c is the side opposite angle C (in this case, PR)
a and b are the lengths of the other two sides (in this case, PQ and QR)
C is the angle opposite side c (in this case, QPR)
Using this formula, we can substitute the given values into the equation:
PR^2 = PQ^2 + QR^2 - 2 * PQ * QR * cos(QPR)
PR^2 = PQ^2 + (14.5cm)^2 - 2 * PQ * (14.5cm) * cos(57°)
Now, we can solve for PR by substituting the given values into the equation and calculating it:
PR^2 = PQ^2 + 210.25cm^2 - 29cm * PQ
Similarly, we can find the length of side PQ by using another application of the Law of Cosines. Rearranging the formula, we have:
PQ^2 = PR^2 + QR^2 - 2 * PR * QR * cos(PQR)
Substituting the given values:
PQ^2 = PR^2 + (14.5cm)^2 - 2 * PR * (14.5cm) * cos(71°)
Again, solve for PQ by substituting the values:
PQ^2 = PR^2 + 210.25cm^2 - 29cm * PR
Solving these two equations will give you the lengths of side PR and PQ.