Fran works due north of home. Her husband Alan works due east. By the time Fran is 7 miles from home, the distance between them is one mile more than Alan's distance from home. How far from home is Alan? Miles

Please and thank you in Advanced!!!

Did you mean "walks" instead of "works" ?

Not that it matters much, I get a right-angled triangle with sides 7, x, and x+1
(x+1)^2 = x^2 + 7^2
x^2 + 2x + 1 = x^2 + 49
2x = 48
x = 24

check: 25^2 = 625
24^2+7^2 = 576+49 = 625

Its works. :)

To solve this problem, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Let's assume that Fran's home is the origin (0,0) on a coordinate plane. Since Fran works due north of home, we can assume that her position is (0,7) (7 miles north of home). Alan works due east of home, so we can assume his position is (x,0) (x miles east of home).

According to the problem statement, the distance between Fran and Alan is one mile more than Alan's distance from home. We can represent this as:

distance between Fran and Alan = Alan's distance from home + 1

Using the distance formula, which states that the distance between two points (x1, y1) and (x2, y2) is given by:

distance = √((x2 - x1)^2 + (y2 - y1)^2)

We can calculate the distance between Fran and Alan as:

distance between Fran and Alan = √((x - 0)^2 + (0 - 7)^2)

Simplifying the equation, we have:

distance between Fran and Alan = √(x^2 + 49)

Since the distance between Fran and Alan is one mile more than Alan's distance from home, we can equate the two expressions:

√(x^2 + 49) = x + 1

To solve for x, we can square both sides of the equation:

x^2 + 49 = (x + 1)^2

Expanding the equation, we get:

x^2 + 49 = x^2 + 2x + 1

Subtracting x^2 from both sides, we have:

49 = 2x + 1

Subtracting 1 from both sides, we get:

48 = 2x

Dividing both sides by 2, we find that:

x = 24

Therefore, Alan is 24 miles from home.

So, Alan is 24 miles from home.