Math Problem Statement

A kite is flying 12 ft off the ground. Its line is pulled taut and casts a 9 ft shadow. Find the length of the line.

Solution

This problem involves using the Pythagorean Theorem to find the length of the kite's line. The scenario described forms a right triangle, with the height and the shadow as the two legs of the triangle, and the kite’s line as the hypotenuse.

  • Height: 12 feet (vertical distance from the ground to the kite)
  • Base: 9 feet (the length of the shadow)
  • Hypotenuse: the kite's line, which we need to find.

The Pythagorean Theorem states:

a2+b2=c2a^2 + b^2 = c^2 where:

  • a=12fta = 12 \, \text{ft} (height),
  • b=9ftb = 9 \, \text{ft} (shadow or base),
  • cc is the length of the kite's line.

Substitute the values into the formula: 122+92=c212^2 + 9^2 = c^2 144+81=c2144 + 81 = c^2 225=c2225 = c^2 Taking the square root of both sides: c=225=15ftc = \sqrt{225} = 15 \, \text{ft}

Therefore, the length of the kite's line is 15 feet.

Would you like further clarification, or do you have any other questions?

Here are some related questions:

  1. How does the Pythagorean Theorem work in real-life applications?
  2. What other methods could be used to solve for the hypotenuse if the triangle was not a right triangle?
  3. How would the solution change if the height or shadow were different?
  4. Could trigonometry help solve this type of problem more efficiently?
  5. How can we apply similar techniques to problems involving three-dimensional geometry?

Tip: Always verify whether a triangle is a right triangle before applying the Pythagorean theorem.

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Math Problem Analysis

Mathematical Concepts

Geometry
Pythagorean Theorem

Formulas

a^2 + b^2 = c^2

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 7-9