Question

Help me find the perimeter please!!

Answer 1

`P = 56\sqrt2`

Explanation:

We can find the perimeter of a square given its diagonal by modeling the square as two congruent right triangles and using the Pythagorean Theorem to solve for the square's side lengths.

The Pythagorean Theorem states that...

`a^2+b^2=c^2`, where
`a` and
`b` are the right triangles legs, and
`c` is its hypotenuse.

However, since we are working with a square, we know that the right triangles' legs are congruent. Thus, the theorem becomes:

`a^2 + a^2 = c^2`

`2 \cdot a^2 = c^2`

Using this formula, we can solve for the square's side lengths, inputting the given hypotenuse length of 28.

`2 \cdot a^2 = 28^2`

↓ simplify the exponent

`2 \cdot a^2 = 784`

↓ divide both sides by 2

`a^2 = 392`

↓ take the square root of both sides

`a = √(392)`

↓ simplify the square root

`a = √(2^3 \cdot 7^2)`

`a = (2\cdot 7)√(2)`

`a=14\sqrt2`

Finally, we can multiply the side length by 4 to get the square's perimeter (since a square has 4 sides).

`P = 4 \cdot a`

`P = 4 \cdot 14\sqrt2`

`\boxed{P = 56\sqrt2}`

`\boxed{P \approx 79.2}`