Question

One of the vertices of Triangle PQR is P(4,-1). The midpoint of Segment PQ is M(6,0). The midpoint of Segment QR is N(8,2). Show that Segment MN is parallel Segment PR and MN = 1/2(PR)

The coordinates of Q are (10,-1)
The coordinates of R are (10,4)
The slope of Segment MN is 1/2
The slope of Segment PR is 1/2
The line segments are parallel
MN = 2
PR = 4

Answer 1

By calculating the slopes of segment MN and PR, we find that both have a slope of 1/2, confirming that MN is parallel to PR. The length of MN is half that of PR as confirmed by midpoint calculations and verified by numeric lengths; MN is 2 units and PR is 4 units.

Step-by-step explanation:

To show that segment MN is parallel to segment PR and MN is half the length of PR, we first identify the coordinates of the vertices and midpoints involved. Vertex P is located at (4, -1), and the midpoints of PQ and QR are M (6, 0) and N (8, 2), respectively. Given the coordinates of Q as (10, -1) and R as (10, 4), we can calculate the slopes.

The slope of MN can be found using the formula for slope, which is the change in y over the change in x. The slope for segment MN is (2 - 0) / (8 - 6) = 1/2.

Moving on to segment PR, using the coordinates of P (4, -1) and R (10, 4), we find the slope to be (4 - (-1)) / (10 - 4) = 5 / 6 = 1/2. Since the slopes are equal, MN is parallel to PR.

To compare lengths, notice that midpoint M is exactly halfway between P and Q, and N is halfway between Q and R. Consequently, the length of MN is half the distance between P and R. Specifically, MN equals the average of MQ and NR, which, considering that Q and R have the same x-coordinate, is halfway between them. Thus, MN is half the length of PR, which can also be observed by looking at the coordinates and verifying MN = 2 and PR = 4.