Question

5) In the figure (ii) given below, AB and CD are two intersecting chords of a circle.
Name two triangles which are similar. Hence, calculate CP given that AP = 6 cm,
PB = 4 cm, and CD
= 14 cm (PC > PD).​

Answer 1

$\underline{\textsf{Given:}}$

$\textsf{AB and CD are two intersecting chords with}$

$\mathsf{AP=6 cm, PB=4 cm\;and\;CD=14 cm}$

$\underline{\textsf{To find:}}$

$\textsf{Length CP}$

$\underline{\textsf{Solution:}}$

$\textsf{Since the chords AB and CD are  intersecting at P, we have}$

$\mathsf{AP{\times}PB=CP{\times}PD}$

$\mathsf{6{\times}4=CP{\times}(CD-CP)}$

$\mathsf{24=CP{\times}(14-CP)}$

$\mathsf{24=14CP-(CP)^2}$

$\mathsf{(CP)^2-14\,CP+24=0}$

$\textsf{Factorise, we get}$

$\mathsf{(CP-2)(CP-12)=0}$

$\implies\mathsf{CP=2,12}$

$\textsf{when CP=2, PD=14-2=12}$

$\textsf{when CP=12, PD=14-12=2}$

$\mathsf{But\;PC\; >\;PD}$

$\therefore\mathsf{CP=12\;cm}$

$\underline{\textsf{Answer:}}$

$\textsf{Length of CP is 12 cm}$