The lengths of the two side of the sails are 5 feet and 12 feet
A boats sail is a right triangle
The diagram is attached below
ABC represents the right angled triangle
AC represents hypotenuse
AC = 13 feet
AB and BC are the other two legs of right angled triangle
Let BC = "a"
So by given length of one side of the sail is 7 feet more than the other side
So AB = a + 7
To find: lengths of the two side of the sails
We have to find length of AB and BC
By pythagoras theorem,
The square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides.
So above definition we can say,
[tex]AC^2 = BC^2 + AB^2[/tex]
Substituting the above values of AB = a + 7 and BC = "a" and AC = 13
[tex]13^2 = a^2 + (a + 7)^2\\\\169 = a^2 + a^2 + 14a + 49\\\\169 = 2a^2 + 14a + 49\\\\2a^2 + 14a - 120 = 0[/tex]
Dividing each term by 2
[tex]a^2 + 7a - 60 = 0[/tex]
Using factoring method to solve this quadratic equation,
"7a" can be written as 12a - 5a
[tex]a^2 + 12a - 5a - 60 = 0[/tex]
Taking "a" as common from first two terms and -5 as common term from next two terms
a(a + 12) -5(a + 12) = 0
Taking (a + 12) as common term,
(a + 12)(a - 5) = 0
Equating each term to zero,
a + 12 = 0 or a - 5 = 0
a = -12 or a = 5
Since length cannot be negative, so ignore a = -12
Thus length of BC = "a" = 5
And length of AB = a + 7 = 5 + 7 = 12
Thus the lengths of the two side of the sails are 5 feet and 12 feet
