Ex. 6.6 Q7 Triangles Solution - NCERT Maths Class 10

Go back to  'Ex.6.6'

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, \(\Delta APC \;\text{and}\; \Delta DPB\)
\[\begin{align} \frac{{AP}}{{PD}}& = \frac{{PC}}{{PB}} = \frac{{AC}}{{DB}}\qquad\quad\left[ {\Delta APC\text{~}\Delta DPB} \right]\\ \frac{{AP}}{{PD}} &= \frac{{PC}}{{PB}} \end{align}\]
\(\Rightarrow \;\;AP.PB = PC.PD\)

  
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