Question

what is the perimeter of △def to the nearest tenth? a 19.4 b 20.1 c 25.3 d 43.3

Explanation:

Step1: Identify triangle type and use trigonometry

$\triangle DEF$ is a right - triangle with $\angle F = 90^{\circ}$, $\angle D=38^{\circ}$ and hypotenuse $DE = 18$. Let's find the lengths of the other two sides $DF$ (adjacent to $\angle D$) and $EF$ (opposite to $\angle D$) using trigonometric ratios.

We know that $\cos D=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{DF}{DE}$ and $\sin D=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{EF}{DE}$.

For $DF$:
$\cos(38^{\circ})=\frac{DF}{18}$
$DF = 18\times\cos(38^{\circ})$
$\cos(38^{\circ})\approx0.7880$, so $DF\approx18\times0.7880 = 14.184$

For $EF$:
$\sin(38^{\circ})=\frac{EF}{18}$
$EF = 18\times\sin(38^{\circ})$
$\sin(38^{\circ})\approx0.6157$, so $EF\approx18\times0.6157 = 11.0826$

Step2: Calculate the perimeter

The perimeter $P$ of a triangle is the sum of its three sides. So $P=DE + DF+EF$

Substitute the values: $DE = 18$, $DF\approx14.184$, $EF\approx11.0826$

$P=18 + 14.184+11.0826=43.2666\approx43.3$

Answer:

D. 43.3