Question

the lighthouse forms a right angle with the path of the boat.
a. write an equation relating h, d, and x.
b. write an indirect proof of the following statement by proving the contrapositive.
if x decreases, then d decreases.
a. an equation relating h, d, and x is d² = h² + x².
b. identify the contrapositive of the given statement.
if d remains the same, then x remains the same.
increases or decreases.
decreases or remains the same.
increases or remains the same.
decreases.
increases.
remains the same.

Explanation:

Step1: Apply Pythagorean theorem

Since it's a right - triangle formed by the lighthouse, the boat's path, and the line from the boat to the top of the lighthouse, we have $d^{2}=h^{2}+x^{2}$.

Step2: Find the contrapositive

The contrapositive of the statement "If $x$ decreases, then $d$ decreases" is "If $d$ does not decrease (i.e., increases or remains the same), then $x$ does not decrease (i.e., increases or remains the same)".

Answer:

a. $d^{2}=h^{2}+x^{2}$
b. If $d$ increases or remains the same, then $x$ increases or remains the same.