Answered

Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Get quick and reliable solutions to your questions from a community of experienced professionals on our platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.

Part A

Prove that when [tex]\( x \ \textgreater \ 1 \)[/tex], a triangle with side lengths [tex]\( a = x^2 - 1 \)[/tex], [tex]\( b = 2x \)[/tex], and [tex]\( c = x^2 + 1 \)[/tex] is a right triangle.

1. Use the Pythagorean theorem:
[tex]\[ a^2 + b^2 = c^2 \][/tex]

2. Substitute the given side lengths into the equation:
[tex]\[ (x^2 - 1)^2 + (2x)^2 = (x^2 + 1)^2 \][/tex]

3. Simplify both sides of the equation:
[tex]\[ (x^2 - 1)^2 = x^4 - 2x^2 + 1 \][/tex]
[tex]\[ (2x)^2 = 4x^2 \][/tex]
[tex]\[ (x^2 + 1)^2 = x^4 + 2x^2 + 1 \][/tex]

4. Combine and simplify:
[tex]\[ x^4 - 2x^2 + 1 + 4x^2 = x^4 + 2x^2 + 1 \][/tex]
[tex]\[ x^4 + 2x^2 + 1 = x^4 + 2x^2 + 1 \][/tex]

5. Conclude that the equation holds true:
[tex]\[ x^4 + 2x^2 + 1 = x^4 + 2x^2 + 1 \][/tex]

Since the equation is valid, the triangle with side lengths [tex]\( a = x^2 - 1 \)[/tex], [tex]\( b = 2x \)[/tex], and [tex]\( c = x^2 + 1 \)[/tex] is a right triangle when [tex]\( x \ \textgreater \ 1 \)[/tex].

Explain your steps:
- Substituted the side lengths into the Pythagorean theorem.
- Simplified and combined terms.
- Verified that both sides of the equation are equal.

Sagot :

GinnyAnswer
Let's solve Part A by proving that the triangle with side lengths [tex]\(a = x^2 - 1\)[/tex], [tex]\(b = 2x\)[/tex], and [tex]\(c = x^2 + 1\)[/tex] is a right triangle when [tex]\(x > 1\)[/tex]. We will use the Pythagorean theorem for this purpose.

The Pythagorean theorem states that for a right triangle with side lengths [tex]\(a\)[/tex], [tex]\(b\)[/tex], and hypotenuse [tex]\(c\)[/tex], the following equation holds:
[tex]\[a^2 + b^2 = c^2.\][/tex]

Using the given side lengths, let's create and simplify this equation step-by-step.

Step 1: Write down the square of each side.

[tex]\[ a^2 = (x^2 - 1)^2 \][/tex]
[tex]\[ b^2 = (2x)^2 \][/tex]
[tex]\[ c^2 = (x^2 + 1)^2 \][/tex]

Step 2: Expand these expressions.

First, expand [tex]\(a^2 = (x^2 - 1)^2\)[/tex]:
[tex]\[ a^2 = (x^2 - 1)(x^2 - 1) \][/tex]
[tex]\[ a^2 = x^4 - 2x^2 + 1 \][/tex]

Next, expand [tex]\(b^2 = (2x)^2\)[/tex]:
[tex]\[ b^2 = 4x^2 \][/tex]

Finally, expand [tex]\(c^2 = (x^2 + 1)^2\)[/tex]:
[tex]\[ c^2 = (x^2 + 1)(x^2 + 1) \][/tex]
[tex]\[ c^2 = x^4 + 2x^2 + 1 \][/tex]

Step 3: Add [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex] to see if it equals [tex]\(c^2\)[/tex].

[tex]\[ a^2 + b^2 = (x^4 - 2x^2 + 1) + 4x^2 \][/tex]
[tex]\[ a^2 + b^2 = x^4 - 2x^2 + 1 + 4x^2 \][/tex]
[tex]\[ a^2 + b^2 = x^4 + 2x^2 + 1 \][/tex]

Step 4: Compare [tex]\(a^2 + b^2\)[/tex] with [tex]\(c^2\)[/tex].

From previous steps:
[tex]\[ a^2 + b^2 = x^4 + 2x^2 + 1 \][/tex]
[tex]\[ c^2 = x^4 + 2x^2 + 1 \][/tex]

We observe that:
[tex]\[ a^2 + b^2 = c^2 \][/tex]

Since the left-hand side of the equation equals the right-hand side of the equation, the given side lengths [tex]\(a = x^2 - 1\)[/tex], [tex]\(b = 2x\)[/tex], and [tex]\(c = x^2 + 1\)[/tex] satisfy the Pythagorean theorem. Hence, the given triangle is a right triangle when [tex]\(x > 1\)[/tex].
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.