Joe is climbing the ladder and stops when his feet are vertically 3.2 feet above the ground and horizontally 2.4 feet from the base of the ladder.
Which equation represents this new situation and can be used to find how far Joe climbed up the ladder?
A. $a^2+2.4^2=3.2^2$
B. $3.2+2.4=c$
C. $3.2^2+b^2=2.4^2$
D. $3.2^2+2.4^2=c^2$
Answer (1)
Recognize the situation as a right triangle.
Apply the Pythagorean theorem: a 2 + b 2 = c 2 .
Substitute the given values: 3. 2 2 + 2. 4 2 = c 2 .
The equation to find how far Joe climbed is: 3. 2 2 + 2. 4 2 = c 2
Explanation
Analyze the problem Joe's position on the ladder, the ground, and the vertical distance from Joe's feet to the ground form a right triangle. The distance Joe's feet are from the base of the ladder horizontally is one leg of the triangle, the vertical distance is the other leg, and the distance Joe climbed up the ladder is the hypotenuse.
Apply the Pythagorean theorem We can use the Pythagorean theorem to find the relationship between these distances. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): a 2 + b 2 = c 2
Substitute the values In this case, a = 3.2 feet (vertical distance) and b = 2.4 feet (horizontal distance). We want to find the equation that represents the distance Joe climbed up the ladder, which is c .
Substituting the given values into the Pythagorean theorem, we get: 3. 2 2 + 2. 4 2 = c 2
State the final equation Therefore, the equation that represents this situation and can be used to find how far Joe climbed up the ladder is: 3. 2 2 + 2. 4 2 = c 2
Examples
The Pythagorean theorem, used here to find the length of the ladder, is also essential in construction and navigation. For example, builders use it to ensure that corners are square, and sailors use it to calculate distances and courses. Understanding this theorem allows us to solve many real-world problems involving right triangles.
