Question

#8 triangles hjk and pmk are similar right triangles. the coordinates of all the vertices are integers. which statement is true of the slope of hk and pk? a. the slope of hk is less than the slope of pk, because the ratio of the change in y - values of the endpoints to the change in x - values of the endpoints is less for hk than it is for pk. b. the slope of hk is greater than the slope of pk, because the ratio of the change in y - values of the endpoints to the change in x - values fo the endpoints is greater for hk than it is for pk. c. the slope of hk is equal to the slope of pk, because the ratio of the change in y - values of the endpoints to the change in x - values of the endpoints is the same for hk as it is for pk. d. the relationship between the slope of hk and the slope of pk cannot be determined, because the triangles are congruent.

Explanation:

Step1: Recall the slope formula

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For similar right - triangles, the ratio of the vertical side (change in \(y\)) to the horizontal side (change in \(x\)) is the same.

Step2: Analyze the slopes of \(HK\) and \(PK\)

Since \(\triangle HJK\sim\triangle PMK\), let the vertical side of \(\triangle HJK\) be \(HJ\) and the horizontal side be \(JK\), and for \(\triangle PMK\) let the vertical side be \(PM\) and the horizontal side be \(MK\).
The slope of a line segment with endpoints \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(m=\frac{\Delta y}{\Delta x}\). For the line segment \(HK\) (using the sides of \(\triangle HJK\)) and \(PK\) (using the sides of \(\triangle PMK\)), because of the similarity of the triangles \(\frac{HJ}{JK}=\frac{PM}{MK}\).
The slope of \(HK\) (calculated using the coordinates of \(H\) and \(K\)) and the slope of \(PK\) (calculated using the coordinates of \(P\) and \(K\)) will have the same ratio of \(\frac{\text{change in }y}{\text{change in }x}\).

Answer:

C. The slope of \(HK\) is equal to the slope of \(PK\), because the ratio of the change in \(y\) - values of the endpoints to the change in \(x\) - values of the endpoints is the same for \(HK\) as it is for \(PK\).