Answer:
The length of q is 7.6 units
The measure of angle R is 31.9°
The measure of angle P is 66.1°
Step-by-step explanation:
Let us use the sine and cosine rules to solve the triangle
In Δ PQR,
p is the opposite side to ∠P
q is the opposite side to ∠Q
r is the opposite side to ∠R
∵ p = 7 units
∵ r = 4 units
∵ m∠Q = 82°
- Angle Q is between p and r so let us find q using cosine rule
∵ q² = p² + r² - 2(p)(r) cos(∠Q)
- Substitute the values of p, r and ∠Q in the rule
∴ q² = (7)² + (4)² - 2(7)(4) cos(82°)
∴ q² = 49 + 16 - 56 cos(82°)
∴ q² = 57.2063
- Take √ for both sides
∴ q = 7.5635
- Round it to the nearest tenth
∴ The length of q is 7.6 units
Now let us use the sine rule to find the measures of ∠R and ∠P
∵ [tex]\frac{q}{sin(Q)}=\frac{r}{sin(R)}=\frac{p}{sin(P)}[/tex]
- Let us find sin(R)
∴ [tex]\frac{7.5635}{sin(82)}=\frac{4}{sin(R)}[/tex]
- By using cross multiplication
∴ 7.5635 × sin(R) = 4 × sin(82)
∴ 7.5635 sin(R) = 4 sin(82)
- Divide both sides 7.5635
∴ sin(R) = 0.5285
- Use [tex]sin^{-1}[/tex] to find m∠R
∵ m∠R = [tex]sin^{-1}[/tex] (0.5285)
∴ m∠R = 31.9042
- Round it to the nearest tenth
∴ The measure of angle R is 31.9°
∵ The sum of the measures of the interior angles of a Δ is 180°
∴ m∠P + m∠Q + m∠R = 180°
∴ m∠P + 82 + 31.9 = 180
∴ m∠P + 113.9 = 180
- Subtract 113.9 from both sides
∴ m∠P = 66.1°
∴ The measure of angle P is 66.1°
