Question

directions
respond to the prompt with at least 100 words. use details about right triangles and the pythagorean theorem as much as possible. not sure how to start? use your response to demonstrate your ability to make connections: text - to - text, text - to - self, and/or text - to - world.
warm - up prompt / materials

• think about a right triangle that has two 45 - degree angles. the two legs are the same length, which are 2 inches long. what is the length of the hypotenuse? simplify completely, without using decimals. what if the sides are 3 inches long? and 5 inches long? how do the side lengths relate to the hypotenuse?
• describe the relationship of the legs to the hypotenuse in your own words. use the numerical values that you found to help develop your ideas.

warm up exercise
question 1
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Explanation:

Step1: Recall Pythagorean theorem

For a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\).

Step2: When \(a = b=2\) inches

Substitute into the Pythagorean theorem: \(c^{2}=2^{2}+2^{2}=4 + 4=8\), so \(c=\sqrt{8}=2\sqrt{2}\) inches.

Step3: When \(a = b = 3\) inches

Substitute: \(c^{2}=3^{2}+3^{2}=9 + 9 = 18\), so \(c=\sqrt{18}=3\sqrt{2}\) inches.

Step4: When \(a = b = 5\) inches

Substitute: \(c^{2}=5^{2}+5^{2}=25+25 = 50\), so \(c=\sqrt{50}=5\sqrt{2}\) inches.

Step5: Describe the relationship

In a right - triangle with two \(45^{\circ}\) angles (an isosceles right - triangle), if the length of each leg is \(x\) inches, the length of the hypotenuse \(c\) is \(x\sqrt{2}\) inches. This is because when \(a = b=x\) in the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), we have \(x^{2}+x^{2}=2x^{2}=c^{2}\), and taking the square root of both sides gives \(c = x\sqrt{2}\).

Answer:

When the legs are 2 inches long, the hypotenuse is \(2\sqrt{2}\) inches. When the legs are 3 inches long, the hypotenuse is \(3\sqrt{2}\) inches. When the legs are 5 inches long, the hypotenuse is \(5\sqrt{2}\) inches. The length of the hypotenuse of an isosceles right - triangle is \(\sqrt{2}\) times the length of each leg.