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
feet.
The distance from Lucas to Maria is
feet.
The angle formed at Maria's position measures
degrees.
To find the distance from Maria to Ava, we can use the sine function:
sin(40°) = Opposite/Hypotenuse
Since the opposite side is the distance from Maria to Ava and the hypotenuse is 12 feet, we have:
sin(40°) = x/12
Cross multiplying, we get:
x = 12 * sin(40°)
Using a calculator, we find that x is approximately 7.7 feet. Therefore, the distance from Maria to Ava is 7.7 feet.
To find the distance from Lucas to Maria, we can use the Pythagorean theorem:
a^2 + b^2 = c^2
Where a and b are the perpendicular sides of the triangle and c is the hypotenuse. In this case, we know that the distance from Ava to Lucas is 12 feet and the distance from Maria to Ava is 7.7 feet. Let's call the distance from Lucas to Maria "d". Plugging the values into the equation, we have:
12^2 + 7.7^2 = d^2
Simplifying, we get:
144 + 59.29 = d^2
203.29 = d^2
Taking the square root of both sides, we find:
d ≈ √203.29
Using a calculator, we find that d is approximately 14.3 feet. Therefore, the distance from Lucas to Maria is 14.3 feet.
Now, to find the angle formed at Maria's position, we can use the sine function again:
sin(angle) = Opposite/Hypotenuse
In this case, the opposite side is the distance from Lucas to Maria (14.3 feet) and the hypotenuse is the distance from Ava to Lucas (12 feet). So:
sin(angle) = 14.3/12
Cross multiplying, we get:
12 * sin(angle) = 14.3
Dividing both sides by 12, we find:
sin(angle) ≈ 1.19
To find the angle, we need to take the inverse sine, also known as the arcsine:
angle ≈ arcsin(1.19)
Using a calculator, we find that the angle is approximately 48.8 degrees. Therefore, the angle formed at Maria's position measures 48.8 degrees.