Question

in δtuv, m∠u = 63°, v = 21 kilometers, and t = 19 kilometers, where v is the length of the side opposite v and t is the length of the side opposite t. find the area of δtuv. write your answer as an integer or as a decimal rounded to the nearest tenth.
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Explanation:

Step1: Recall the formula for the area of a triangle with two sides and included angle

The formula for the area \( A \) of a triangle when two sides \( a \) and \( b \) and the included angle \( C \) are known is \( A=\frac{1}{2}ab\sin C \). In \( \triangle TUV \), we know sides \( t = 19 \) (opposite \( T \)) and \( v=21 \) (opposite \( V \)), and the included angle \( \angle U = 63^\circ \). So we can use the formula \( A=\frac{1}{2} \times t\times v\times\sin(\angle U) \).

Step2: Substitute the given values into the formula

Substitute \( t = 19 \), \( v = 21 \), and \( \angle U=63^\circ \) into the formula:
\( A=\frac{1}{2}\times19\times21\times\sin(63^\circ) \)

First, calculate \( 19\times21 = 399 \). Then, \( \frac{1}{2}\times399=199.5 \). Now, find \( \sin(63^\circ) \approx 0.8910 \).

Then, multiply \( 199.5\times0.8910 \approx 199.5\times0.891 \)

\( 199.5\times0.891=(200 - 0.5)\times0.891=200\times0.891-0.5\times0.891=178.2 - 0.4455 = 177.7545 \)

Rounding to the nearest tenth, we get \( 177.8 \)

Answer:

\( 177.8 \)