Answer:
Explanation:
To find the perimeter of a right-angled triangle when only the length of the hypotenuse is known, you'll need to use the Pythagorean theorem. Here's a step-by-step explanation:
Step 1: Understand the Pythagorean theorem
The Pythagorean theorem 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):
c^2 = a^2 + b^2
Step 2: Substitute the known length of the hypotenuse
Let's say the length of the hypotenuse is "h". Substitute "h" for "c" in the Pythagorean theorem:
h^2 = a^2 + b^2
Step 3: Solve for one of the unknown sides
To find the perimeter of the triangle, we need to know the lengths of both sides "a" and "b". To do this, we need to solve for one of the unknown sides. Let's solve for "a".
Square root both sides of the equation:
√(h^2) = √(a^2 + b^2)
h = √(a^2 + b^2)
Square both sides of the equation:
h^2 = a^2 + b^2
Step 4: Use the Pythagorean theorem to find the other unknown side
Now that we have an expression for "a", we can use the Pythagorean theorem again to find "b":
h^2 = a^2 + b^2
Substitute the expression for "a" that we found in Step 3 into the equation:
h^2 = (√(a^2 + b^2))^2 + b^2
h^2 = a^2 + b^2 + b^2
h^2 = 2 * b^2 + a^2
h^2 - a^2 = 2 * b^2
b^2 = (h^2 - a^2) / 2
Take the square root of both sides:
b = √((h^2 - a^2) / 2)
Step 5: Calculate the perimeter
The perimeter of the triangle is the sum of the lengths of all three sides, so we can now calculate the perimeter using the lengths of "a" and "b" that we found in Steps 3 and 4:
P = a + b + h
Step 6: Conclusion
Therefore, the perimeter of the right-angled triangle can be found by first using the Pythagorean theorem to find the lengths of two of the sides, and then adding up the lengths of all three sides to find the perimeter.