Question

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$$\begin{aligned} &\\text{triangle 1: sides } 8, 6, x \\\\ &a^2 + b^2 = c^2 \\\\ &6^2 + 8^2 = x^2 \\\\ &36 + 64 = x^2 \\\\ &100 = x^2 \\\\ &x = \\sqrt{100} \\\\ &x = 10 \\\\ \\\\ &\\text{triangle 2: sides } 4, 3, x \\\\ &a^2 + b^2 = c^2 \\\\ &3^2 + 4^2 = x^2 \\\\ &9 + 16 = x^2 \\\\ &25 = x^2 \\\\ &x = \\sqrt{25} \\\\ &x = 5 \\end{aligned}$$

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Explanation:

Identify the error in the first triangle calculation

The first triangle has side lengths \(8\) and \(6\), and hypotenuse \(x\).
The student wrote:

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$$

The student wrote \(x = 10\) but their final line shows a handwritten error where they wrote \(x = 10\) but it looks like \(16\) or they miscalculated the square root of \(100\) as \(10\) (which is correct, but written unclearly as \(10\)). Let's verify the math:

$$ \sqrt{100} = 10 $$

The student's written work shows \(x = 10\) (the digit \(0\) has a slight vertical line, making it look like \(10\)).

Identify the second triangle calculation

The second triangle has legs \(3\) and \(4\), and hypotenuse \(x\).
The student wrote:

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$$

This calculation is correct:

$$ \sqrt{25} = 5 $$

Summarize the solutions for both problems

We will present the correct step-by-step application of the Pythagorean theorem for both triangles shown in the image.

Answer:

Question 1

For the first triangle with legs \(a = 8\) and \(b = 6\):

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Question 2

For the second triangle with legs \(a = 3\) and \(b = 4\):

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$$