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 ch
consider the distance in the hypothesis of the contrapositive. if it increases, then the ec must increase.
if the distance in the hypothesis decreases, then the distance in the conclusion must d
if the distance in the hypothesis remains the same, then the distance in the conclusion

Explanation:

Step1: Recall Pythagorean theorem

Since the lighthouse forms a right - angle with the path of the boat, by the Pythagorean theorem, for a right - triangle with sides \(h\), \(x\) and hypotenuse \(d\), 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 \(d\) increases or remains the same, then \(x\) increases or remains the same". Given \(d^{2}=h^{2}+x^{2}\), where \(h\) is fixed. If \(d\) remains the same, then from \(d^{2}=h^{2}+x^{2}\), we have \(x^{2}=d^{2}-h^{2}\). Since \(d\) and \(h\) are constant, \(x\) must remain 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. When \(d\) remains the same, \(x\) must remain the same.