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 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 must increase. if the distance in the hypothesis decreases, then the distance in the conclusion must decrease. if the distance in the hypothesis remains the same, then the distance in the conclusion must remain the same. each possibility for the contrapositives hypothesis leads to the conclusion being. it follows that the contrapositive is

Explanation:

Step1: Find the relation 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: Identify 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 or remains the same, $x$ must increase or remain the same. So the contrapositive 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 true.