Math Problem Statement
A straight length of wire is 20 cm long. It is bent at right angles to form the two shorter sides of a right angled traignle. If hte triangle's area is 30cm^2, find: i. the length of the hypotenuse ii. the triangle's perimeter
Solution
We are given a wire that is 20 cm long and is bent at right angles to form the two shorter sides of a right-angled triangle. The area of the triangle is 30 cm².
Let:
- and be the two shorter sides (legs) of the triangle.
- The total length of the wire is cm.
- The area of the triangle is given as 30 cm², so cm².
Step 1: Express one side in terms of the other.
From , we can solve for one variable:
Step 2: Substitute into the area formula.
Using , we substitute into the equation: Multiply both sides by 2: Expand the equation: Rearrange into a standard quadratic form:
Step 3: Solve the quadratic equation.
We will solve using the quadratic formula: Simplifying the square root: Thus, we have two possible solutions for :
Since both values satisfy the equation , we can proceed to find the hypotenuse using these approximate values of and .
Let me know how to proceed to find the
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Math Problem Analysis
Mathematical Concepts
Geometry
Algebra
Quadratic Equations
Formulas
Area of a right-angled triangle: (1/2)ab
Perimeter of a triangle: a + b + c
Quadratic equation formula: ax^2 + bx + c = 0
Pythagorean theorem: a^2 + b^2 = c^2
Theorems
Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a
Pythagorean Theorem
Suitable Grade Level
Grades 9-10
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