Question

Rex, Paulo, and Ben are standing on 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 and the dolphin, and Ben and the dolphin.

URGENT !! I NEED TO PASS THIS TO GRADUATE !!

Answer 1

Given information:
`\triangle ADC\sim \triangle ACB` and
`\triangle BDC\sim \triangle BCA`.

To Prove :
`a^2+b^2=c^2`

Proof:

Statement Proof

1.
`\triangle ADC\sim \triangle ACB` 1. Given

2.
`(AC)/(AB)=(AD)/(AC)` 2. The ratio of corresponding parts of similar triangles is a constant

3.
`(b)/(c)=(e)/(b)` 3. Rewrite statement 2 using given side labels.

4.
`b^2=ce` 4. Cross multiply statement 4

5.
`\triangle BDC\sim \triangle BCA` 5. Given

6.
`(BC)/(BA)=(BD)/(BC)` 6. The ratio of corresponding parts of similar triangles is a constant

7.
`(a)/(c)=(d)/(a)` 7. Rewrite statement 5 using given side labels.

8.
`a^2=cd` 8. Cross multiply statement 7

9.
`a^2+b^2=cd+ce` 9. Add statement 4 and 8

10.
`a^2+b^2=c(d+e)` 10. c is common factor

11.
`a^2+b^2=c^2` 11. Segment addition property.

Hence proved.

Answer 2

1.
`m\angle BAC=m\angle CAD,\ m\angle ACB=m\angle ADC=90^(\circ)`, then
`m\angle ABC=m\angle ACD` and triangles ADC and ACB are similar by AAA theorem.

2. The ratio of the corresponding sides of similar triangles is constant, so

`(AC)/(AB)= (AD)/(AC)`.

3. Knowing lengths you could state that
`(b)/(c)= (e)/(b)`.

4. This ratio is equivalent to
`b^2=ce`.

5.
`m\angle ABC=m\angle CBD,\ m\angle ACB=m\angle CDB=90^(\circ)`, then
`m\angle BAC=m\angle BCD` and triangles BDC and BCA are similar by AAA theorem.

6. The ratio of the corresponding sides of similar triangles is constant, so

`(BC)/(BD)= (AB)/(BC)`.

7. Knowing lengths you could state that
`(a)/(d)= (c)/(a)`.

8. This ratio is equivalent to
`a^2=cd`.

9. Now add results of parts 4 and 8:

`b^2+a^2=ce+cd`.

10. c is common factor, then:

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

11. Since
`e+d=c` you have
`a^2+b^2=c\cdot c=c^2`.