# Ex. 6.6 Q7 Triangles Solution - NCERT Maths Class 10

## Question

In Fig. below, two chords \(AB\) and \(CD\) intersect each other at the point \(P. \)

**Prove that:**

(i) \(\Delta APC\,\text{~}\Delta {\text{ }}DPB\)

(ii) \(AP. PB = CP. DP \)

## Text Solution

**Reasoning:**

As we know that, two triangles, are similar if:

(i) Their corresponding angles are equal.

(ii) Their corresponding sides are in the same ratio.

As we know that angles in the same segment of a circle are equal.

**Steps:**

(i) In, \(\Delta APC \;\text{and}\; \Delta DPB\)

\(\angle APC = \angle DPB\) (Vertically opposite angles)

\(\angle PAC = \angle PDB\) (Angles in the same segment)

\(\Rightarrow \Delta APC\text{~}\Delta DPB\) (A.A criterion)

(ii) In, \(\Delta APC \;\text{and}\; \Delta DPB\)

\[\begin{align} \frac{{AP}}{{PD}}& = \frac{{PC}}{{PB}} = \frac{{AC}}{{DB}}\qquad\quad\left[ {\Delta APC\text{~}\Delta DPB} \right]\\ \frac{{AP}}{{PD}} &= \frac{{PC}}{{PB}} \end{align}\]

\(\Rightarrow \;\;AP.PB = PC.PD\)