A right triangle has a hypotenuse of length 13 meters and one side of length 5 meters. What is the length of the third side and the angle between the hypotenuse and the 5 meter side?

A. 144 meters, 67.38 degrees
B. 12 meters, 22.62 degrees
C. 10 meters, 22.62 degrees
D. 12 meters, 67.38 degrees

I know the length is 12 meters but I am having difficulty finding the degrees, Please Help!

D. 12m, 67.38 degrees

To find the length of the third side and the angle between the hypotenuse and the 5 meter side, we can use the Pythagorean theorem and trigonometric functions.

Let's label the sides of the right triangle:
Hypotenuse: c = 13 meters
Side: a = 5 meters
Third side: b (unknown)

Using the Pythagorean theorem, we know that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides:

c^2 = a^2 + b^2

Substituting the known values:
13^2 = 5^2 + b^2
169 = 25 + b^2
b^2 = 144
b = √144
b = 12 meters

So the length of the third side is 12 meters.

To find the angle between the hypotenuse and the 5 meter side, we can use the sine function:

sin(angle) = opposite/hypotenuse

In this case, the side opposite the angle is 5 meters and the hypotenuse is 13 meters.

sin(angle) = 5/13

Using an inverse sine function (arcsine) on both sides, we can find the angle:

angle = arcsin(5/13)
angle ≈ 22.62 degrees

Therefore, the length of the third side is 12 meters and the angle between the hypotenuse and the 5 meter side is approximately 22.62 degrees.

So the correct answer is option B) 12 meters, 22.62 degrees.

To find the length of the third side and the angle between the hypotenuse and the 5-meter side, we can use the Pythagorean theorem and trigonometric functions.

1. Using the Pythagorean theorem:
According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In this case, we have:
Hypotenuse^2 = side1^2 + side2^2
13^2 = 5^2 + side2^2
169 = 25 + side2^2
side2^2 = 169 - 25
side2^2 = 144

Taking the square root of both sides, we get:
side2 = √144
side2 = 12 meters

So, the length of the third side is 12 meters.

2. To find the angle between the hypotenuse and the 5-meter side:
We can use the trigonometric function sine.

In this case, we know the opposite side (5 meters) and the hypotenuse (13 meters). The sine of an angle is equal to the ratio of the length of the opposite side to the length of the hypotenuse.

sin(angle) = opposite / hypotenuse
sin(angle) = 5 / 13

Now, we need to find the angle whose sine is equal to 5/13. We can use the inverse sine or arcsine function (sin^-1) to do this.

angle = sin^-1(5/13)
angle ≈ 22.62 degrees

Therefore, the correct answer is option B: 12 meters, 22.62 degrees.