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:

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<reasoning>

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
\[

$$\begin{aligned} a &= 2(3) = 6 \\ b &= 2(3) + 2 = 8 \\ c &= 2(3) + 4 = 10 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(a = 6, b = 8, c = 10.\)</mcq-correct>
<mcq-option>(B) \(a = 10, b = 8, c = 26.\)</mcq-option>
</answer>

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Answer:

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<reasoning>

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
\[

$$\begin{aligned} a &= 2(3) = 6 \\ b &= 2(3) + 2 = 8 \\ c &= 2(3) + 4 = 10 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(a = 6, b = 8, c = 10.\)</mcq-correct>
<mcq-option>(B) \(a = 10, b = 8, c = 26.\)</mcq-option>
</answer>

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