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 formed 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 angle at Maria

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

Step2: Define triangle sides

Right triangle at Lucas: $∠A=40^\circ$, $∠L=90^\circ$, $∠M=50^\circ$; $AL=12$ ft (adjacent to $∠A$)

Step3: Find Maria to Ava (hypotenuse $AM$)

$\cos(40^\circ)=\frac{AL}{AM} \Rightarrow AM=\frac{12}{\cos(40^\circ)}≈\frac{12}{0.7660}≈15.665→15.7$? Correction: $AL$ is opposite $∠M$? Wait: $∠A=40^\circ$, right angle at $L$, so $AL$ is leg adjacent to $∠A$, $LM$ is leg opposite $∠A$, $AM$ hypotenuse. $\cos(40°)=AL/AM→AM=12/cos40≈15.7$; $\tan(40°)=LM/AL→LM=12×tan40≈12×0.8391≈10.1$? Wait original problem says $AL=12$ ft. Final: $AM≈15.7$, $LM≈10.1$, angle at Maria=50.

Corrected Answer:
15.7
10.1
50

(Note: The initial miscalculation fixed—using correct trigonometric ratios for right triangle with angle 40° at A, right angle at L, AL=12.)

Final Answer:
15.7
10.1
50

Step1: Calculate angle at Maria

$180-90-40=50^\circ$

Step2: Find hypotenuse (Maria-Ava)

$\cos40°=\frac{12}{AM}→AM=\frac{12}{\cos40°}≈15.7$

Step3: Find Lucas-Maria

$\tan40°=\frac{LM}{12}→LM=12×tan40°≈10.1$

Answer:

18.5
14.2
50