A child’s kite is on top of a straight pine tree. A ladder is placed against the tree and touches the kite. The ladder forms a 45-degree angle with the flat ground. The base of the tree is 9 feet away from the base of the ladder. With only this information, how can you determine the height of the tree?
This problem involves a right isosles triangle.
Legs of an isosles triangle have the same measure.
To determine the height of the tree, we can use trigonometry. Let's start by drawing a right triangle to represent the situation. The ladder represents the hypotenuse of the triangle, while the height of the tree represents one of the legs. The base of the ladder represents the other leg.
Since the ladder forms a 45-degree angle with the flat ground, we know that this is a special right triangle called an isosceles right triangle. In an isosceles right triangle, the two legs are equal in length, which means that the length of the ladder is also the length of the base of the tree.
Based on the given information, the base of the ladder is 9 feet. Therefore, we have the length of one leg (9 feet) and the measure of the angle between this leg and the hypotenuse (45 degrees).
Now, we can use the trigonometric function called sine (sin) to find the height of the tree. In this case, we want to find the sin of the angle, which is equal to the ratio of the length of the opposite side (the height of the tree) to the length of the hypotenuse (the ladder).
Using the formula sin(angle) = opposite/hypotenuse, we can plug in the values we know: sin(45 degrees) = height/9 feet.
Since the sine of 45 degrees is equal to √2/2 (approximately 0.7071), we can solve the equation for the height of the tree:
0.7071 = height/9 feet
To find the height, we can multiply both sides of the equation by 9:
0.7071 * 9 feet = height
This gives us an approximate height of the tree:
6.36 feet ≈ height of the tree
Therefore, based on the given information, the height of the tree is approximately 6.36 feet.