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9. a) how far does mrs. owen have to walk to get from the parking lot to the beach?

9. b) how far does she have to walk to get from the parking lot to the refreshment stand?

Let's call the distance from the beach to the playground x.
Let's call the distance from the stand to the parking lot z.
Let's call the base of the biggest triangle in the picture (from the parking lot to the left) y.

Recall the pythagorean theorem: the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse.

There are three right triangles we can work with.

Triangle with sides 48, x and z. Here z is the hypotenuse so we get the equation

Triangle with sides 27, x and y. Here y is the hypotenuse so we get

Triangle with sides z, y and (48+27). Here (48+27) is the hypotenuse so we get . That is,

We have 3 equations and 3 unknowns (x, y and z). So let's take the last equation: and replace and using expressions that contain an expression in terms of x from the first two equations we came up with.

That gives us: which we solve for x as follows:



Now that we know x we can substitute 36 for x into the equation and solve for z as follows:


This means that the distance from the beach to the parking lot is 36 m and the distance from the stand to the parking lot is 60 m.