Math Problem Statement
Solution
Let's define the variables:
- Let be the length of the shorter leg.
- The length of the longer leg is given as feet.
- The length of the hypotenuse is feet.
We can apply the Pythagorean theorem to solve for :
Expanding both sides:
Substituting these into the equation:
Simplifying:
Move all terms to one side:
This simplifies to:
Solving this quadratic equation:
Using the quadratic formula:
Thus, the possible solutions are:
Since the length cannot be negative, .
Final side lengths:
- Shorter leg feet.
- Longer leg feet.
- Hypotenuse feet.
Would you like further clarification or a detailed explanation of any step?
Here are 5 related questions to expand on this topic:
- How does the Pythagorean theorem apply to different types of triangles?
- What are the possible methods to solve quadratic equations?
- Can you derive a formula for the hypotenuse in any right triangle?
- What are some real-life applications of the Pythagorean theorem?
- How would the equation change if the triangle were isosceles?
Tip: Always check the discriminant () when solving quadratic equations to see if real solutions exist.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Pythagorean Theorem
Formulas
Pythagorean theorem: a^2 + b^2 = c^2
Quadratic equation formula: ax^2 + bx + c = 0
Quadratic formula: x = [-b ± sqrt(b^2 - 4ac)] / 2a
Theorems
Pythagorean Theorem
Suitable Grade Level
Grades 8-10
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