Find the smallest side of triangle PQR.

Steps to Solve

1. Find the Third Angle

The sum of the angles in a triangle is always 180°.

To find the third angle $A$, use the equation: $$ A = 180° - (56° + 61°) $$

Calculating gives: $$ A = 180° - 117° = 63° $$

2. Identify the Corresponding Sides

The sides opposite the angles can be determined:

• Angle $A = 63°$ is opposite side $a$ (PQ).
• Angle $B = 56°$ is opposite side $b$ (QR).
• Angle $C = 61°$ is opposite side $c$ (RP).

3. Use the Law of Sines

The Law of Sines states that: $$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$

This can be rearranged to find the sides: $$ a = k \sin A, \quad b = k \sin B, \quad c = k \sin C $$ for some constant $k$.

4. Determine the Size of Each Side

Now calculate the ratios:

• For side $a$: $$ a = k \sin(63°) $$

• For side $b$: $$ b = k \sin(56°) $$

• For side $c$: $$ c = k \sin(61°) $$

5. Compare the Sides

We can compare the values of $\sin$ for the angles:

• $\sin(63°)$
• $\sin(56°)$
• $\sin(61°)$

Since the side length is directly proportional to the sine of the angle, the side opposite the smallest angle (56°) will be the smallest side.

6. Conclusion on Smallest Side

The smallest angle is 56°, so the smallest side is $b = QR$.