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 ava’s 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 toy’s path. give each distance in feet rounded to the nearest tenths place.
the distance from maria to ava is \boxed{} feet.
the distance from lucas to maria is \boxed{} feet.
the angle formed at maria’s position measures \boxed{} degrees.

Explanation:

Step1: Find angle at Maria

Sum of angles in triangle: $180^\circ - 90^\circ - 40^\circ = 50^\circ$

Step2: Distance Maria to Ava (hypotenuse)

Use cosine: $\cos(40^\circ) = \frac{12}{c} \Rightarrow c = \frac{12}{\cos(40^\circ)} \approx 15.7$

Step3: Distance Lucas to Maria

Use tangent: $\tan(40^\circ) = \frac{b}{12} \Rightarrow b = 12 \tan(40^\circ) \approx 9.7$

Answer:

15.7, 9.7, 50