Question

what is the approximate measure of angle t in the triangle below?
t
2.7 cm
s
3.9 cm
4.3 cm
u
not drawn to scale
79°
84°
96°
101°

Explanation:

Step1: Identify the sides and the Law of Cosines

We have triangle \( STU \) with sides \( ST = 2.7 \) cm, \( TU = 3.9 \) cm, and \( SU = 4.3 \) cm. We want to find angle \( T \). The Law of Cosines states that for a triangle with sides \( a \), \( b \), \( c \) and the angle opposite side \( c \) being \( C \), \( c^{2}=a^{2}+b^{2}-2ab\cos(C) \). Here, side \( SU \) is opposite angle \( T \), so \( SU^{2}=ST^{2}+TU^{2}-2\cdot ST\cdot TU\cdot\cos(T) \).

Step2: Plug in the values

Substitute \( SU = 4.3 \), \( ST = 2.7 \), \( TU = 3.9 \) into the formula:
\[
4.3^{2}=2.7^{2}+3.9^{2}-2\times2.7\times3.9\times\cos(T)
\]
Calculate the squares: \( 4.3^{2}=18.49 \), \( 2.7^{2} = 7.29 \), \( 3.9^{2}=15.21 \)
\[
18.49=7.29 + 15.21-2\times2.7\times3.9\times\cos(T)
\]
Simplify the right - hand side: \( 7.29+15.21 = 22.5 \), and \( 2\times2.7\times3.9=21.06 \)
\[
18.49=22.5-21.06\cos(T)
\]

Step3: Solve for \( \cos(T) \)

Rearrange the equation:
\[
21.06\cos(T)=22.5 - 18.49
\]
\[
21.06\cos(T)=4.01
\]
\[
\cos(T)=\frac{4.01}{21.06}\approx0.1904
\]

Step4: Find the angle \( T \)

Take the inverse cosine: \( T=\cos^{- 1}(0.1904)\approx101^{\circ} \)

Answer:

\( 101^{\circ} \)