A 24-foot-tall tree casts a shadow that is 10 feet long. What is the distance from the top of the tree to the top of its shadow? Apply a Pythagorean triple to solve this problem
To solve this problem, we will use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Let's consider the height of the tree as one side of the right triangle (let's call it side a) and the length of the shadow as another side (let's call it side b). We are trying to find the distance from the top of the tree to the top of its shadow, which represents the hypotenuse (let's call it side c).
According to the Pythagorean theorem, a^2 + b^2 = c^2.
In this case, side a is 24 feet (height of the tree), and side b is 10 feet (length of the shadow). We want to find side c.
Plugging in the values, we get:
24^2 + 10^2 = c^2
Simplifying, we have:
576 + 100 = c^2
676 = c^2
Taking the square root of both sides, we get:
c = √676
c = 26
Therefore, the distance from the top of the tree to the top of its shadow is 26 feet.
To apply a Pythagorean triple to solve this problem, we notice that the numbers 24, 10, and 26 form a Pythagorean triple (24^2 + 10^2 = 26^2). In this case, the triple is a multiple of the 3-4-5 Pythagorean triple (3*8, 3*2, 3*10), where 24 is a multiple of 3 and 10 is a multiple of 2.