Question

Triangle DEF contains right angle E. If angle D measures 55° and its opposite side measures 7.6 units, what is the measure of side

DF? Round your answer to the nearest hundredth.

A. 5.34 units
B. 6.38 units
C. 9.06 units
D. 9.28 units

Answer 1

D. 9.28 units

Explanation:

In a right triangle, the side opposite the right angle is called the hypotenuse. So, in right triangle DEF, where ∠E is the right angle, side DF is the hypotenuse. (See attachment).

Given that we know the angle opposite side DF, along with the measurement of another angle and its corresponding opposite side, we can apply the Sine Rule to determine the length of side DF.

`\boxed{\begin{array}{l}\underline{\textsf{Sine Rule}} \\\\(\sin A)/(a)=(\sin B)/(b)=(\sin C)/(c)\\\\\textsf{where:}\\\phantom{ww}\bullet \;\textsf{$A, B$ and $C$ are the angles.}\\\phantom{ww}\bullet\;\textsf{$a, b$ and $c$ are the sides opposite the angles.}\end{array}}`

In this case:

`(\sin D)/(d)=(\sin E)/(e)=(\sin F)/(f)`

Given:

• E = 90°
• D = 55°
• d = EF = 7.6
• e = DF

Substitute the values into the Sine Rule formula:

`(\sin 55^(\circ))/(7.6)=(\sin 90^(\circ))/(DF)`

Cross-multiply:

`DF\sin 55^(\circ)=7.6\sin 90^(\circ)`

Divide both sides by sin 55°:

`DF=(7.6\sin 90^(\circ))/(\sin 55^(\circ))`

Calculate:

`DF=(7.6(1))/(0.819152044...)`

`DF=(7.6)/(0.819152044...)`

`DF=9.277886874...`

`DF=9.28\; \sf units\;(nearest\;hundredth)`

Therefore, the measure of side DF is 9.28 units, rounded to the nearest hundredth.