Math Problem Statement
Solution
The problem describes a right triangle with an altitude of 20 cm. The goal is to express the length of the hypotenuse as a function of the perimeter of the triangle.
Let’s break down the steps:
Variables:
- The right triangle has two legs and a hypotenuse:
- One leg is the altitude = 20 cm.
- Let the other leg be .
- The hypotenuse is .
The perimeter of the triangle is the sum of the three sides:
Using the Pythagorean theorem, we can express the relation between the sides of the right triangle:
Now, solve for in terms of . From the perimeter equation:
Substitute this expression for into the Pythagorean theorem:
This equation gives as a function of . However, solving this equation for explicitly in terms of involves algebraic manipulation and potentially solving a quadratic equation.
Would you like to see the steps to simplify this further? Let me know if you have any questions!
Related Questions:
- How can you simplify the expression ?
- What is the significance of the Pythagorean theorem in right triangles?
- How does the altitude affect the perimeter of the triangle?
- How would the function change if the altitude was different?
- Can this approach be used for triangles with different altitudes?
Tip: Always check for extraneous solutions when working with square roots in equations.
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Math Problem Analysis
Mathematical Concepts
Algebra
Geometry
Pythagorean Theorem
Formulas
Perimeter: p = 20 + x + h
Pythagorean Theorem: h^2 = 400 + x^2
Hypotenuse in terms of perimeter: h = sqrt(400 + (p - 20 - h)^2)
Theorems
Pythagorean Theorem
Suitable Grade Level
Grades 9-12
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