Question - Demonstration of the Pythagorean Theorem
Solution:
The image shows an equation representing the Pythagorean Theorem which states that in a right-angled triangle, the square of the length of the hypotenuse $$(c)$$ is equal to the sum of the squares of the lengths of the other two sides $$(a)$$ and $$(b)$$. The Pythagorean Theorem is expressed as:\[ a^2 + b^2 = c^2 \]
Tutor Solution:
The Pythagorean Theorem is a fundamental principle in geometry that relates the lengths of the sides of a right-angled triangle. The theorem is expressed by the formula $$a^2 + b^2 = c^2$$, where $$a$$ and $$b$$ are the lengths of the legs of the triangle, and $$c$$ is the length of the hypotenuse, which is the side opposite the right angle.Here's a step-by-step explanation of the concept:1. Identify a right-angled triangle. A right-angled triangle is one in which one of the angles measures exactly 90 degrees.2. Label the sides of the triangle. The two sides that form the right angle are referred to as the legs of the triangle and are labeled $$a$$ and $$b$$. The longest side, which is opposite the right angle, is known as the hypotenuse and is labeled $$c$$.3. Apply the Pythagorean Theorem. According to the theorem, the square of the length of the hypotenuse ($$c^2$$) is equal to the sum of the squares of the lengths of the other two sides ($$a^2$$ and $$b^2$$).4. For example, if you know the lengths of the two legs of a right-angled triangle, you can find the length of the hypotenuse by rearranging the theorem to solve for $$c$$: $$c = \sqrt{a^2 + b^2}$$.5. Conversely, if you know the length of the hypotenuse and one of the legs, you can find the length of the other leg by rearranging the theorem to solve for $$a$$ or $$b$$: $$a = \sqrt{c^2 - b^2}$$ or $$b = \sqrt{c^2 - a^2}$$.The Pythagorean Theorem is widely used in various applications, including mathematics, engineering, architecture, and physics among others. It allows for the calculation of distances and is essential in trigonometry. This theorem only applies to right-angled triangles and is a cornerstone for understanding properties of these types of triangles.
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