Question

type the correct answer in each box. use numerals instead of words.
ava, lucas, and maria are playing a game where they pass a pull - back toy to one another in a path that forms a right triangle. ava sends the toy to lucas. then, lucas sends the toy to maria. finally, maria sends it back to ava. the angle formed at lucas position measures 90° and the angle formed at avas position measures 40°. the distance from ava to lucas is 12 feet.
ava is trying to determine the other measures of the triangle created by the toys path. give each distance in feet rounded to the nearest tenths place.
the distance from maria to ava is . feet.
the distance from lucas to maria is feet.
the angle formed at marias position measures degrees.

Explanation:

Step1: Find the angle at Maria's position

In a triangle, the sum of angles is \(180^{\circ}\). Let the angle at Maria's position be \(x\).
We know one angle is \(90^{\circ}\) (at Lucas) and one is \(40^{\circ}\) (at Ava). So \(x = 180-(90 + 40)=50^{\circ}\)

Step2: Find the distance from Maria to Ava (hypotenuse \(c\))

We know the adjacent side \(a = 12\) (Ava - Lucas) and the angle at Ava \(\theta=40^{\circ}\). Using the cosine formula \(\cos\theta=\frac{a}{c}\), where \(a = 12\), \(\theta = 40^{\circ}\)
\(c=\frac{12}{\cos(40^{\circ})}\)
Since \(\cos(40^{\circ})\approx0.766\), \(c=\frac{12}{0.766}\approx15.7\)

Step3: Find the distance from Lucas to Maria (opposite side \(b\))

Using the tangent formula \(\tan\theta=\frac{b}{a}\), where \(a = 12\), \(\theta = 40^{\circ}\)
\(b=a\times\tan(40^{\circ})\)
Since \(\tan(40^{\circ})\approx0.839\), \(b = 12\times0.839\approx10.1\)

Answer:

The distance from Maria to Ava is \(15.7\) feet.
The distance from Lucas to Maria is \(10.1\) feet.
The angle formed at Maria's position measures \(50\) degrees.