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.

infinitely many solutions

step-by-step explanation:

we have been given the equation;

2(2x + 3) = -4 + 2(2x + 5)

we need to determine the value of x.

2(2x + 3), we expand this expression to obtain 4x + 6

similarly 2(2x + 5) will become 4x + 10

now we have;

4x + 6 = -4 + 4x + 10

4x + 6 = 4x + 10 - 4

4x + 6 = 4x + 6

the expression on the left hand side is identical to the one on the right hand side. therefore, we are looking at just a single straight line whose equation is;

y = 4x + 6

consequently, we shall have infinitely many solutions