Question - Solving for x in a Right Triangle using the Pythagorean Theorem
Solution:
The image shows a right triangle with one of the angles marked as 90 degrees, indicating that it's indeed a right triangle. There are two sides labeled: one is labeled "10" and the other is labeled "2x". Since the side labeled "10" is opposite the right angle, it is the hypotenuse.To solve for "x," you can use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.The theorem is written as:$$ a^2 + b^2 = c^2 $$where $$ c $$ is the hypotenuse, and $$ a $$ and $$ b $$ are the other two sides.From the image, we have:$$ a = 2x $$$$ c = 10 $$We are missing the length of the other side, but since the triangle's sides are marked with the same hash mark, it means they are of equal length. Therefore, $$ b = 2x $$ as well.Now we can insert the values into the Pythagorean theorem:$$ (2x)^2 + (2x)^2 = 10^2 $$Expanding this gives us:$$ 4x^2 + 4x^2 = 100 $$Combine like terms:$$ 8x^2 = 100 $$Divide both sides by 8 to solve for $$ x^2 $$:$$ x^2 = \frac{100}{8} $$$$ x^2 = 12.5 $$Taking the square root of both sides gives us $$ x $$:$$ x = \sqrt{12.5} $$$$ x \approx 3.54 $$ (to two decimal places).So, $$ x $$ is approximately 3.54.
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