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 Maria's angle

Sum of triangle angles is $180^\circ$.
$180^\circ - 90^\circ - 40^\circ = 50^\circ$

Step2: Find Lucas to Maria distance

Use tangent: $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$
$\tan(40^\circ)=\frac{\text{Lucas-Maria}}{12}$
$\text{Lucas-Maria}=12\times\tan(40^\circ)\approx12\times0.8391=10.1$

Step3: Find Maria to Ava distance

Use cosine: $\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$
$\cos(40^\circ)=\frac{12}{\text{Maria-Ava}}$
$\text{Maria-Ava}=\frac{12}{\cos(40^\circ)}\approx\frac{12}{0.7660}=15.7$

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.