Question

In the rectangle ABCD, AB = 9 cm, BC = 12 cm. Midpoints M and N of AB and BC are joined. Length of MN is ______.

Options

• 4.5 cm

• 6 cm

• 7.5 cm

• 8 cm

Answer 1

In the rectangle ABCD, AB = 9 cm, BC = 12 cm. Midpoints M and N of AB and BC are joined. Length of MN is 7.5 cm.

Explanation:

Given:

• ABCD is a rectangle, with:
  • AB = 9 cm 
  • BC = 12 cm
• M and N are the midpoints of sides AB and BC, respectively.

We need to find the length of segment MN, the line joining the midpoints M and N.

Step 1: Apply the Midpoint Theorem

The Midpoint Theorem states that the segment joining the midpoints of two sides of a triangle is parallel to the third side and is half the length of that third side. Although the situation here is in a rectangle, we can apply the theorem by treating the triangle formed by the points A, B and C (since M and N are midpoints of the sides of the rectangle).

In the rectangle, the segment MN joins the midpoints of sides AB and BC, so MN is parallel to the diagonal AC and half its length.

Step 2: Find the Length of AC

Using the Pythagorean theorem in triangle ABC, we calculate the length of diagonal AC:

`AC = sqrt(AB^2 + BC^2)`

= `sqrt(9^2 + 12^2)`

= `sqrt(81 + 144)`

= `sqrt(225)`

= 15 cm

Step 3: Find the Length of MN

Since MN is half the length of AC (by the Midpoint Theorem), we have:

`MN = 1/2 xx AC`

= `1/2 xx 15`

= 7.5 cm