Question - Solving for the Length of a Side in a Right-Angled Triangle
Solution:
The image shows a right-angled triangle, which means we can use the Pythagorean theorem to solve for the missing side, labeled as "x." The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.In the given triangle, we have one side that is 9 cm and another side that is 12 cm. Assuming that the 12 cm side is the hypotenuse (since it's opposite what is presumably the right angle), we can compute the length of side "x" using the following equation:$$ c^2 = a^2 + b^2 $$Here, $$ c $$ represents the hypotenuse, while $$ a $$ and $$ b $$ represent the other two sides. Rearranging the equation to solve for $$ a $$, we get:$$ a^2 = c^2 - b^2 $$Let's substitute the given values:$$ x^2 = 12^2 - 9^2 $$$$ x^2 = 144 - 81 $$$$ x^2 = 63 $$Taking the square root of both sides to solve for $$ x $$, we find:$$ x = \sqrt{63} $$The square root of 63 can be simplified further by factoring out perfect squares. $$ 63 = 9 \times 7 $$, and since $$ 9 $$ is a perfect square, we get:$$ x = \sqrt{9 \times 7} $$$$ x = \sqrt{9} \times \sqrt{7} $$$$ x = 3\sqrt{7} $$Therefore, the length of side $$ x $$ is $$ 3\sqrt{7} $$ cm, which cannot be simplified further without a decimal approximation. If a decimal approximation is needed, $$ \sqrt{7} \approx 2.645 $$, and multiplying this by 3 gives:$$ x \approx 3 \times 2.645 $$$$ x \approx 7.935 $$So the side "x" is approximately 7.935 cm long.
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