Question

Rex, Paulo, and Ben are standing on the shore watching for dolphins. Paulo sees one surface directly in front of him about a hundred feet away. Use the spaces provided below to prove that the square of the distance between Rex and Ben is the same as the sum of the squares of the distances between Rex and the dolphin, and Ben and the dolphin.

Answer 1

I just need help on how to solve the proof

Answer 2

In general, there is a theorem that would be useful here and perhaps allow the skipping of a couple steps (geometric mean leg theorem if I'm not mistaken). But the proof of that theorem relies on similar triangles, which is what is given in this problem anyway.

Answer 3

I know number one is given

Answer 4

For number 2 would it be AC/BC=BC/AC?

Answer 5

The first reason would be "Given." We are given this information in the problem.

The second statement would be:

$$\frac{AC}{AB}=\frac{AD}{AC}$$

Answer 6

ah ok

Answer 7

The reason for this is that AC is the hypotenuse of the small triangle. So it corresponds to AB, which is the hypotenuse of the biggest triangle

Answer 8

Now, AC is also the smallest side of the biggest triangle, so we want to find the smallest side of the small triangle (ADC). This is AD.

Answer 9

so then the third would be b/c=e/b

Answer 10

Yes. Now cross-multiply.

$$\frac{b}{c}=\frac{e}{b}$$

So b^2 = ce

Answer 11

Cross multiply would be the reason for number four.

Five is complete.

Six is going to be similar to two.

Answer 12

would it be BC/BA=BD/BC

Answer 13

For six, it may help to look at seven and work backwards.

Yes, you're right. The ratio of the longer leg to the hypotenuse in the biggest triangle equals the ratio of the longer leg to hypotenuse in the second-biggest triangle.

Answer 14

So the reason for six would be the same as that for two.

The reason for seven will be the same as that for three.

In eight, cross-multiply to get that a^2=cd

Answer 15

Now, if we add the equation in statement four to that in statement eight, we get:

$$a^2+b^2=cd+ce$$

Answer 16

What would be for number 10 then?

Answer 17

Now factor out c on the right side.

a^2+b^2 = c(d+e)

But d + e is c.

So a^2 + b^2 = c*c = c^2

Answer 18

The factoring would be number 10.

Answer 19

which is the Pythagorean theorem THANKS!