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 $d^{2}=h^{2}+x^{2}$.
b. identify the contrapositive of the given statement
if d increases or remains the same, then x increases or remains the same.
of the three distances, only d and x can change. h is/are fixed and cannot change.
consider the distance in the hypothesis of the contrapositive. if it increases, then the equation from part a indicates the distance in the conclusion of the contrapositive

Explanation:

Step1: Establish the equation for part a

Since it's a right - triangle situation (lighthouse forms a right - angle with the boat's path), by the Pythagorean theorem, the equation relating the height of the lighthouse $h$, the distance from the boat to the base of the lighthouse $x$, and the slant - distance from the top of the lighthouse to the boat $d$ is $d^{2}=h^{2}+x^{2}$.

Step2: Find the contrapositive for part b

The contrapositive of the statement "If $x$ decreases, then $d$ decreases" is "If $d$ increases or remains the same, then $x$ increases or remains the same".

Step3: Prove the contrapositive

We know from the equation $d^{2}=h^{2}+x^{2}$, where $h$ is fixed. If $d$ increases or remains the same, then $d^{2}$ increases or remains the same. Since $d^{2}=h^{2}+x^{2}$ and $h^{2}$ is constant, for $d^{2}$ to increase or remain the same, $x^{2}$ (and thus $x$) must increase or remain the same.

Answer:

a. $d^{2}=h^{2}+x^{2}$
b. The contrapositive "If $d$ increases or remains the same, then $x$ increases or remains the same" is proved as above.