Question

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$$\begin{aligned} &b^2 = c^2 - a^2 \\\\ &12^2 + x^2 = 16^2 \\\\ &144 + x^2 = 256 \\\\ &x^2 = 256 - 144 \\\\ &x^2 = 122 \\\\ &x = 10 \\end{aligned}$$

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Explanation:

Identify the given values

We have a right triangle.
The vertical leg is \(12\).
The horizontal leg is \(x\).
The hypotenuse is \(16\).

Set up the equation

Using the Pythagorean Theorem knowledge point

$$ 12^2 + x^2 = 16^2 $$

Simplify the terms

Using the Pythagorean Theorem knowledge point

$$ 144 + x^2 = 256 $$

Solve for the unknown

Using the Pythagorean Theorem knowledge point

$$
LATEXBLOCK0
$$

Calculate the final value

Using the Square Roots knowledge point

$$
LATEXBLOCK1
$$

Answer:

Based on the right triangle shown in the image, the correct calculation to find the value of \(x\) is:

$$12^2 + x^2 = 16^2$$

$$144 + x^2 = 256$$

$$x^2 = 256 - 144$$

$$x^2 = 112$$

$$x = \sqrt{112} = 4\sqrt{7} \approx 10.58$$

(Note: The handwritten work in the image contains subtraction errors, stating \(256 - 144 = 122\) instead of \(112\), and incorrectly simplifying \(\sqrt{122}\) to \(10\).)

The correct value of \(x\) is <blank>\(4\sqrt{7}\)</blank>.