Question

The points M and N lie respectively on the sides AB and BC of ΔABC , with AM : MB = BN : NC

= 2 : 1 . The points E and F are the midpoints of AB and BC respectively. Prove that E, F and the midpoint MN are
collinear.​

Answer 1

To prove that E, F, and the midpoint MN are collinear, we can use the line segment approach. By calculating the coordinates and slopes of the points, we can show that the slope between E and MN is 0, indicating collinearity.

Step-by-step explanation:

To prove that points E, F, and the midpoint MN are collinear, we can use the line segment approach.

First, let's label the coordinates of the points. Let A be (0, 0), B be (r, 0), and C be (2r, 0), where r is the length of AB and BC.

Since E is the midpoint of AB, its coordinates are (r/2, 0). Similarly, the coordinates of F, the midpoint of BC, are (3r/2, 0).

To find the coordinates of the midpoint MN, we can use the ratio between the lengths AM:MB = 2:1. This means that the x-coordinate of M is r/3, and the x-coordinate of N is 2r/3. The y-coordinate of both M and N is 0, since they lie on the x-axis.

Now, let's calculate the slope between E and MN. The slope between two points (x1, y1) and (x2, y2) is given by (y2 - y1) / (x2 - x1).

In this case, the slope between E and MN is (0 - 0) / (r/3 - r/2) = 0.

Since the slope is 0, this means that the points E, F, and the midpoint MN are collinear.