Fill in the blanks so that the following statements are true:

□ ABCD is trapezium in which ||. The diagonals intersect in P. If PD = 9, AP = 5, PB = 7.2, then AC = ______.

We have

Given: ABCD is a trapezium.

AD ∥ BC

PD = 9,

AP = 5 &

PB = 7.2

To find: AC = ?

We have got the trapezium ABCD, in which

AD ∥ BC & P is the intersection point of AC and BD.

Take ∆PAD and ∆PCB,

∠APD = ∠CPB [∵, Vertically opposite angles are equal]

∠PAD = ∠PCB [∵, alternate angles]

∠PDA = ∠PBC [∵, alternate angles]

⇒ By AAA similarity theorem, ∆PAD ∼ ∆PCB for the correspondence PAD ↔ PCB.

Now by definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.

⇒

Reciprocating it, we get

⇒

⇒

⇒ PC = 4

From the diagram, note that

AC = AP + PC

⇒ AC = 5 + 4

⇒ AC = 9

Thus, the answer is 9.