Question

How do I solve this problem?

Answer 1

The length of a square is the same on all sides.

From the Pythagorean theorum we know that the length of a diagonal (C) is equal to the square root of the sum of the sides (A and B) squared:


`{a}^(2) + {b}^(2) = {c}^(2)`

We also know that the area of a square is equal to the length of one of the sides squared.


`{l}^(2) = a`

so to find the length of the diagonal we first need to find the length of a side. we do this using the area of the square Equation:


`{l}^(2) = 196 \\ l = 14`

now we plug 14 into the Pythagorean theorum. because it's a square we know that a=b.


`l = 14 = a = b`

plugging 14 into Pythagorean theorum:


`{a}^(2) + {b}^(2) = {c}^(2) \\ {14}^(2) + {14}^(2) = {c}^(2) \\ 196 + 196 = {c}^(2) \\ 392 = {c}^(2) \\ c = √(392)`

simplify the √392 by breaking it up into factors and simplifying the largest perfect square.

1*392=392
2*196
4*98
7*56
8*49
14*28=392

These are all the numbers that multiply together to make equal 392 are called factors

Find one that has a perfect square. Here there are two.
4*98 and 8*49

pick one to simplify. it doesn't matter which. I'll choose 8*49


`√(392) = √(8 * 49) \\ √(8 * 49) = \sqrt{8 * {7}^(2) } \\ \sqrt{8 * {7}^(2) } = 7 √(8)`

Notice this isn't one of our answers. That's because we can do this process again with the √8:


`7 √(8) = 7 √(4 * 2 ) \\ 7 √(4 * 2 ) = 7 \sqrt{ {2}^(2) * 2 } \\ 7 \sqrt{ {2}^(2) * 2 } = (7 * 2) √(2) \\ = 14 √(2)`

So C is the answer.

Answer 2

The diagonal is 14sqrt(2) ft

Explanation:

The area of a square is given by

A =s^2 where s is the side length

196 = s^2

Take the square root of each side

sqrt(196) =sqrt(s^2)

14 = s

The side length is 14 ft

Now we can use the Pythagorean theorem to find the length of the diagonal

a^2 + b^2 = c^2

The side lengths of the triangle are s = 14, and the diagonal is the hypotenuse

14^2 + 14^2 = c^2

196+196 = c^2

392 = c^2

Take the square root of each side

sqrt(392) = sqrt(c^2)

sqrt(196 *2) = c

We can separate this into two pieces sqrt(ab) = sqrt(a) sqrt(b)

sqrt(196)sqrt(2) =c

14sqrt(2) = c

The diagonal is 14sqrt(2) ft