Question

ABCD is a square. P divides AB in the ratio 4:1 and Q divides CD in the ratio 3:2. M is the midpoint of AD. Prove that PQ bisects MC.

a) True
b) False

Answer 1

To prove that PQ bisects MC, we need to show that the line segment PQ divides the line segment MC into two equal parts. By using the ratios given, we can conclude that PQ does indeed bisect MC.

Step-by-step explanation:

To prove that PQ bisects MC, we need to show that the line segment PQ divides the line segment MC into two equal parts.

Since P divides AB in the ratio 4:1, we can say that AP:PB = 4:1. Similarly, since Q divides CD in the ratio 3:2, we can say that CQ:QD = 3:2.

Now, let's consider triangles MCQ and MCP. Using the properties of similar triangles, we can conclude that MQ:MP = CQ:CP. Since CQ:CP = 3:4 (from the ratios given), we can deduce that MQ:MP = 3:4. This shows that PQ bisects MC, as the ratio of the lengths is equal on both sides of the dividing line.