Question

find the sides of the given triangle.

\\(b = 2x + 2\\)
\\(c = 2x + 4\\)
\\(a = 2x\\)

\\(a = 6, b = 8, c = 10.\\)
\\(a = 10, b = 8, c = 26.\\)

Explanation:

Set up the Pythagorean equation

Using the Quadratic Equations knowledge point

$$ a^2 + b^2 = c^2 $$

$$ (2x)^2 + (2x + 2)^2 = (2x + 4)^2 $$

Expand and simplify the algebraic terms

Using the Quadratic Equations knowledge point

$$ 4x^2 + (4x^2 + 8x + 4) = 4x^2 + 16x + 16 $$

$$ 8x^2 + 8x + 4 = 4x^2 + 16x + 16 $$

Rearrange into standard quadratic form

Using the Quadratic Equations knowledge point

$$ 4x^2 - 8x - 12 = 0 $$

$$ x^2 - 2x - 3 = 0 $$

Solve for x by factoring

Using the Quadratic Equations knowledge point

$$ (x - 3)(x + 1) = 0 $$

$$ x = 3 \quad (\text{since } x > 0 \text{ for positive side lengths}) $$

Calculate the side lengths

Using the Linear Equations knowledge point

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Answer:

• (A) \(a = 6, b = 8, c = 10.\) (Correct answer)
• (B) \(a = 10, b = 8, c = 26.\)