The length of side r is 12.9 meters.
The triangle in the image is a right triangle, because one of its angles is a right angle. This means that we can use the Pythagorean Theorem to find the length of side r. The Pythagorean Theorem states that 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, the hypotenuse is side r, and the other two sides are q and s. Therefore, we can write the following equation:
r² = q² + s²
We are given that q = 9 meters, and we can find the length of side s using trigonometry. We know that the angle opposite side s is m<Q = 42°, and we are given that the hypotenuse is side r. Therefore, we can use the sine function to find the length of side s:
sin(42°) = s / r
Solving for s, we get:
s = r * sin(42°)
Now we can substitute this expression for s in the Pythagorean Theorem equation:
r² = q² + (r * sin(42°))²
Distributing the parentheses on the right side, we get:
r² = q² + r² * sin²(42°)
Subtracting r² from both sides of the equation, we get:
r² - r² * sin²(42°) = q²
Dividing both sides of the equation by 1 - sin²(42°), we get:
r² = q² / (1 - sin²(42°))
Taking the square root of both sides, we get:
r = √(q² / (1 - sin²(42°)))
Substituting q = 9 meters and m<Q = 42° into the equation, we get:
r = √((9 meters)² / (1 - sin²(42°)))
Evaluating the expression, we get:
r = 12.9 meters
Therefore, the length of side r is 12.9 meters.