At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Discover in-depth solutions to your questions from a wide range of experts on our user-friendly Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
Certainly! Let's solve the problem step-by-step.
Step 1: Identify the type of triangle
A triangle with angles [tex]\(45^\circ\)[/tex], [tex]\(45^\circ\)[/tex], and [tex]\(90^\circ\)[/tex] is a right-angled isosceles triangle. This means it has two equal sides opposite the two [tex]\(45^\circ\)[/tex] angles.
Step 2: Represent the sides of the triangle
Let each of the equal sides (legs) of the triangle be represented by [tex]\(a\)[/tex]. The hypotenuse (the side opposite the [tex]\(90^\circ\)[/tex] angle) will be represented by [tex]\(c\)[/tex].
Step 3: Use the Pythagorean theorem
In a right-angled triangle, the Pythagorean theorem states:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
For this specific triangle, since both legs are of equal length ([tex]\(a = b\)[/tex]), the theorem simplifies to:
[tex]\[ a^2 + a^2 = c^2 \][/tex]
Step 4: Combine like terms
Simplifying the left-hand side of the equation:
[tex]\[ 2a^2 = c^2 \][/tex]
Step 5: Solve for [tex]\(c\)[/tex]
To find the length of the hypotenuse, [tex]\(c\)[/tex], solve the equation for [tex]\(c\)[/tex] by taking the square root of both sides:
[tex]\[ c = \sqrt{2a^2} \][/tex]
[tex]\[ c = \sqrt{2} \cdot \sqrt{a^2} \][/tex]
Since [tex]\(\sqrt{a^2} = a\)[/tex]:
[tex]\[ c = \sqrt{2} \cdot a \][/tex]
Conclusion:
The length of the hypotenuse [tex]\(c\)[/tex] is [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\(a\)[/tex]. Thus, we have shown that:
[tex]\[ c = \sqrt{2} \cdot a \][/tex]
This verifies that the length of the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as each leg.
Step 1: Identify the type of triangle
A triangle with angles [tex]\(45^\circ\)[/tex], [tex]\(45^\circ\)[/tex], and [tex]\(90^\circ\)[/tex] is a right-angled isosceles triangle. This means it has two equal sides opposite the two [tex]\(45^\circ\)[/tex] angles.
Step 2: Represent the sides of the triangle
Let each of the equal sides (legs) of the triangle be represented by [tex]\(a\)[/tex]. The hypotenuse (the side opposite the [tex]\(90^\circ\)[/tex] angle) will be represented by [tex]\(c\)[/tex].
Step 3: Use the Pythagorean theorem
In a right-angled triangle, the Pythagorean theorem states:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
For this specific triangle, since both legs are of equal length ([tex]\(a = b\)[/tex]), the theorem simplifies to:
[tex]\[ a^2 + a^2 = c^2 \][/tex]
Step 4: Combine like terms
Simplifying the left-hand side of the equation:
[tex]\[ 2a^2 = c^2 \][/tex]
Step 5: Solve for [tex]\(c\)[/tex]
To find the length of the hypotenuse, [tex]\(c\)[/tex], solve the equation for [tex]\(c\)[/tex] by taking the square root of both sides:
[tex]\[ c = \sqrt{2a^2} \][/tex]
[tex]\[ c = \sqrt{2} \cdot \sqrt{a^2} \][/tex]
Since [tex]\(\sqrt{a^2} = a\)[/tex]:
[tex]\[ c = \sqrt{2} \cdot a \][/tex]
Conclusion:
The length of the hypotenuse [tex]\(c\)[/tex] is [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\(a\)[/tex]. Thus, we have shown that:
[tex]\[ c = \sqrt{2} \cdot a \][/tex]
This verifies that the length of the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as each leg.
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.