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
must increase.
if the distance in the hypothesis decreases, then the distance
both be true.
if the distance in the hypothesis remains the same, then the d
have opposite truth values
the same.
each possibility for the contrapositives hypothesis leads to th
have the same truth value
it the contrapositive is true.
for any statement, the statement and its contrapositive must
therefore, the given statement is true, completing the proof.

Explanation:

Step1: Establish right - triangle equation

Since the lighthouse forms a right - angle with the path of the boat, by the Pythagorean theorem, we have $d^{2}=h^{2}+x^{2}$.

Step2: Find the contrapositive

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

Step3: Analyze the contrapositive

Given $d^{2}=h^{2}+x^{2}$, where $h$ is fixed. If $d$ increases or remains the same, from the equation $x^{2}=d^{2}-h^{2}$, when $d$ increases, $d^{2}$ increases (since $h^{2}$ is constant), so $x^{2}$ increases and $x$ increases; when $d$ remains the same, $x$ also remains the same. Since the contrapositive is true, the original statement is true.

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 proven to be true, so the original statement "If $x$ decreases, then $d$ decreases" is true.