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 square of the distances between rex and the dolphin, and ben and the dolphin.

given: δabc ~ δacb and δbdc ~ δbca
prove:

Please look at the steps below.

Step-by-step explanation:

1) m∠BAC=m∠CAD,   m∠ACB=m∠ADC=90 degrees , then  m<ABC=m∠ACD.

Therefore, triangles ADC and ACB are similary by AAA theorem.

2)AC /AB = AD/AC

3) Substituting the lengths, we get b/c = e/b

4) Cross multiplying the ratio, we get b^2 = ce

5)  m∠ABC=m∠CBD, m∠ACB=m∠CDB=9, then  m∠BAC=m∠BCD  and the triangles BDC and BCA are similar by AAA theorem.

6)  The ratio of the corresponding sides of similar triangles is in proportion.

BC/BD = AB/BC

7) Substituting the lengths, we get a/d = c/a

8) a^2 = cd

9) Add the results in 4) and 8), we get b^2 + a^2 = ce + cd

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

11) e + d =c, when we plug in step 10, we get  a^2 + b^2 = c.c

a^2 + b^2 = c^2.

Hope this will helpful.

Thank you.

2. angles qrt and qst both measures 156 degrees.

step-by-step explanation:  

we have been given a diagram of a circle and we are told that measure of arc qvt is 156 degrees.  

we can see from our given diagram that angle qrt and angle qst subtends to arc qvt.

since we know that angles subtended by the same arc at the circumference are always equal.

by angle subtended by same arc theorem, measure of angle qrt and angle qst will be equal to the measure of arc qvt.

therefore, angles qrt and qst both measures 156 degrees and 2nd statement is the correct choice.