Question

Nautical flags are used to represent letters of alphabets. The flag for the letter O consists of a yellow right triangle and a red right triangle joined together along their hypotenuse to form a square. The joint hypotenuse of the two triangles is three inches longer than a side of the square. Find the length of a side of the flag. Round your answer to the nearest tenth.

Answer 1

7.2 in

Explanation:

Let length of side of flag=x

Hypotenuse of right triangle=x+3

According to question information

`(x+3)^2=x^2+x^2`

Using Pythagoras theorem

`(hypotenuse)^2=(base)^2+(perpendicular\;side)^2`

`x^2+6x+9=2x^2`

Using identity:
`(x+y)^2=x^2+y^2+2xy`

`2x^2-x^2-6x-9=0`

`x^2-6x-9=0`

Using quadratic formula :
`x=(-b\pm√(b^2-4ac))/(2a)`

`x=(6\pm√((-6)^2-4(1)(-9)))/(2(1))`

`x=(6\pm√(36+36))/(2)`

`x=(6\pm√(72))/(2)`

`x=(6\pm6\sqrt2)/(2)`

`x=(6+6\sqrt2)/(2)=3+3\sqrt2=7.2`

`x=(6-6\sqrt2)/(2)=3-3\sqrt2`

`x=-1.24`

It is not possible because the length of side is always positive.

Hence, the side of flag=7.2 in

Answer 2

The length of a side of the flag is 7.2inches

Explanation:

Let length = x

Let hypotenuse = x+3

Using Pythagoras' Theorem:

x^2 + x^2 = (x+3)^2

x^2 + x^2 = x^2 + 6x + 9

2x^2 = x^2 + 6x + 9

×^2 - 6x - 9 = 0

`x = \frac{ - b + - \sqrt{ {b}^(2) - 4ac } }{2a}`

x=[-(-6)+sqrt.(-6)^2-4(1)(-9)]/2(1)

×=[6+sqrt.72]/2

×=7.2inches

×=[-(-6)+sqrt.(-6)^2-4(1)(-9)]/2(1)

x=[6-sqrt.72]/2

×=-1.2(rej)

(PS. sqrt is square root)

(Correct me if i am wrong)