For each of the following, tell how many noncongruent triangles PQR fit the given description, and find the size of angle Q. Make a separate diagram for each case.
(a) p = 3, q = 5, angle P = 27 degrees (b) p = 8, q = 5, angle P = 57 degrees
(c) p = 7, q = 8, angle P = 70 degrees (d) p = 10, q = 20, angle P =30 degrees
To find the number of noncongruent triangles PQR and the size of angle Q for each case, 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:
sin A / a = sin B / b = sin C / c
Let's break down each case:
(a) p = 3, q = 5, angle P = 27 degrees:
We know two sides (p and q) and their opposite angle (P). To find the number of noncongruent triangles, we need to consider the possible lengths of the third side, r.
To do this, we can use the Law of Sines:
sin P / p = sin Q / q
Substituting the given values, we have:
sin 27 / 3 = sin Q / 5
We can solve for sin Q by multiplying both sides by 5:
sin Q = 5 * (sin 27 / 3) ≈ 0.722
To find the value of Q, we can take the inverse sine (sin^(-1)) of 0.722:
Q ≈ sin^(-1)(0.722) ≈ 45.3 degrees
You can draw a diagram for this case, using the given values and the calculated angle measures.
(b) p = 8, q = 5, angle P = 57 degrees:
We follow the same approach as in case (a):
sin P / p = sin Q / q
sin 57 / 8 = sin Q / 5
Solving for sin Q:
sin Q = 5 * (sin 57 / 8) ≈ 0.543
Taking the inverse sine:
Q ≈ sin^(-1)(0.543) ≈ 33.1 degrees
Create a diagram for this case, representing the given values and the calculated angle measures.
(c) p = 7, q = 8, angle P = 70 degrees:
Again, using the Law of Sines:
sin 70 / 7 = sin Q / 8
sin Q = 8 * (sin 70 / 7) ≈ 0.876
Taking the inverse sine:
Q ≈ sin^(-1)(0.876) ≈ 60.3 degrees
Draw a diagram for this case, including the given values and the calculated angle measures.
(d) p = 10, q = 20, angle P = 30 degrees:
Applying the Law of Sines:
sin 30 / 10 = sin Q / 20
sin Q = 20 * (sin 30 / 10) = 20 * (1/2) = 10
Here, we notice that sin Q equals 1, which is the maximum possible value for a sine function. This means that angle Q can be any angle between 0 and 90 degrees.
You can draw a diagram for this case, representing the given values and the range of possible angle measures for Q.