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

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

$\dfrac{AC}{AB}= \dfrac{AD}{AC}$.

3. Knowing lengths you could state that $\dfrac{b}{c}= \dfrac{e}{b}$.

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

$\dfrac{BC}{BD}= \dfrac{AB}{BC}$.

7. Knowing lengths you could state that $\dfrac{a}{d}= \dfrac{c}{a}$.

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 $a^2+b^2=c\cdot c=c^2$.

Answer 2

Answer:

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. $\dfrac{AC}{AB}=\dfrac{AD}{AC}$                                     2. The ratio of corresponding parts of similar triangles is a constant

3. $\dfrac{b}{c}=\dfrac{e}{b}$                                           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.  $\dfrac{BC}{BA}=\dfrac{BD}{BC}$                                   6. The ratio of corresponding parts of similar triangles is a constant

7. $\dfrac{a}{c}=\dfrac{d}{a}$                                            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.