3. 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
Oh, triangles! I love a good geometry challenge. Let's tri this out!
(a) For p = 3, q = 5, and angle P = 27 degrees, we can have only one noncongruent triangle PQR. And for angle Q? Well, since angle P and angle Q must add up to 180 degrees in a triangle, angle Q would be 180 - 27 = 153 degrees. Voila!
(b) Now, for p = 8, q = 5, and angle P = 57 degrees, we can have two noncongruent triangles PQR. One angle P is 57 degrees, but for the second triangle, angle P would be the supplementary angle, which is 180 - 57 = 123 degrees. As for angle Q, it would be 180 - angle P, so 180 - 57 = 123 degrees again. Two triangles, double the fun!
(c) With p = 7, q = 8, and angle P = 70 degrees, we can have only one noncongruent triangle PQR. And for angle Q? It would be 180 - 70 = 110 degrees. That's all we need!
(d) Lastly, for p = 10, q = 20, and angle P = 30 degrees, we can have two noncongruent triangles PQR. Angle P is 30 degrees for the first triangle, but for the second triangle, it would be the supplementary angle, 180 - 30 = 150 degrees. And angle Q? It would be 180 - angle P, so 180 - 30 = 150 degrees. Double the triangles, double the laughs!
I hope my answers tri-angled your curiosity! If you need more geometry humor, just give me a shout.
To determine the number of noncongruent triangles PQR and find the size of angle Q, we can use the Triangle Inequality Theorem and the Law of Sines.
First, let's draw the diagrams for each case:
(a) p = 3, q = 5, angle P = 27 degrees:
Q
/ \
p / \ q
/___\
R P
(b) p = 8, q = 5, angle P = 57 degrees:
Q
/ \
p / \ q
/___\
R P
(c) p = 7, q = 8, angle P = 70 degrees:
Q
/ \
p / \ q
/___\
R P
(d) p = 10, q = 20, angle P = 30 degrees:
Q
/ \
p / \ q
/___\
R P
To solve each case, we'll use the Law of Sines, which states that the ratio of the sine of an angle in a triangle to the length of its opposite side is constant:
sin(P)/p = sin(Q)/q
Note: In this case, we will assume the angle is measured in degrees.
(a) For case (a), we have sin(27)/3 = sin(Q)/5
Rearranging the equation, we get sin(Q) = (sin(27)/3) * 5
Using a scientific calculator, we find sin(Q) ≈ 0.454
Therefore, angle Q ≈ arcsin(0.454) ≈ 27.6 degrees
(b) For case (b), we have sin(57)/8 = sin(Q)/5
Rearranging the equation, we get sin(Q) = (sin(57)/8) * 5
Using a scientific calculator, we find sin(Q) ≈ 0.784
Therefore, angle Q ≈ arcsin(0.784) ≈ 52.1 degrees
(c) For case (c), we have sin(70)/7 = sin(Q)/8
Rearranging the equation, we get sin(Q) = (sin(70)/7) * 8
Using a scientific calculator, we find sin(Q) ≈ 0.79
Therefore, angle Q ≈ arcsin(0.79) ≈ 52.6 degrees
(d) For case (d), we have sin(30)/10 = sin(Q)/20
Rearranging the equation, we get sin(Q) = (sin(30)/10) * 20
Using a scientific calculator, we find sin(Q) ≈ 1
Therefore, angle Q ≈ arcsin(1) ≈ 90 degrees
Hence, we have solved each case and found the size of angle Q for the given descriptions.