In the given figure, two chords $AB$ and $CD$ intersect each other at the point $P$. Prove that
(i) $∆APC \sim ∆DPB$.
(ii) $AP \times PB = CP \times DP$.
"
Given:
Two chords $AB$ and $CD$ intersect each other at the point $P$.
To do:
We have to prove that
(i) $∆APC \sim ∆DPB$.
(ii) $AP \times PB = CP \times DP$.
Solution:
(i) In $\triangle \mathrm{APC}$ and $\triangle \mathrm{BPD}$,
$\angle \mathrm{APC}=\angle \mathrm{DPB}$ (Vertically opposite angles)
$\angle \mathrm{PAC}=\angle \mathrm{PDB}$ (Angles on the same segment are equal)
Therefore, by AA similarity,
$\triangle \mathrm{APC} \sim \triangle \mathrm{DPB}$
Hence proved.
(ii) In $\triangle \mathrm{APC}$ and $\triangle \mathrm{BPD}$,
$\angle \mathrm{APC}=\angle \mathrm{DPB}$ (Vertically opposite angles)
$\angle \mathrm{PAC}=\angle \mathrm{PDB}$ (Angles on the same segment are equal)
Therefore, by AA similarity,
$\triangle \mathrm{APC} \sim \triangle \mathrm{DPB}$ 
This implies,
$\frac{\mathrm{AP}}{\mathrm{PD}}=\frac{\mathrm{CP}}{\mathrm{PB}}$
$\mathrm{AP} \times \mathrm{PB}=\mathrm{CD} \times \mathrm{PD}$
Hence proved.
Related Articles
- In the given figure, two chords $AB$ and $CD$ intersect each other at the point $P$. Prove that $AP \times PB = CP \times DP$."
- In the given figure, two chords $AB$ and $CD$ of a circle intersect each other at the point $P$ (when produced) outside the circle. Prove that(i) $∆PAC \sim ∆PDB$.(ii) $PA \times PB = PC \times PD$."
- In the given figure, altitudes $AD$ and $CE$ of $∆ABC$ intersect each other at the point $P$. Show that:(i) $∆AEP \sim ∆CDP$(ii) $∆ABD \sim ∆CBE$(iii) $∆AEP \sim ∆ADB$(iv) $∆PDC \sim ∆BEC$"
- In the given figure, two chords $AB$ and $CD$ of a circle intersect each other at the point $P$ (when produced) outside the circle. Prove that $PA \times PB = PC \times PD$."
- In the given figure, altitudes $AD$ and $CE$ of $∆ABC$ intersect each other at the point $P$. Show that:$∆AEP \sim ∆ADB$"
- In the given figure, altitudes $AD$ and $CE$ of $∆ABC$ intersect each other at the point $P$. Show that $∆PDC \sim ∆BEC$"
- In the given figure, altitudes $AD$ and $CE$ of $∆ABC$ intersect each other at the point $P$. Show that:"
- In the given figure, if $∆ABE \cong ∆ACD$, show that $∆ADE \sim ∆ABC$."
- In the given figure, $D$ is a point on hypotenuse $AC$ of $∆ABC, DM \perp BC$ and $DN \perp AB$. Prove that:(i) $DM^2 = DN \times MC$(ii) $DN^2 = DM \times AN$"
- In the given figure, $D$ is a point on side $BC$ of $∆ABC$, such that $\frac{BD}{CD}=\frac{AB}{AC}$. Prove that $AD$ is the bisector of $∆BAC$."
- In the given figure, $D$ is a point on hypotenuse $AC$ of $∆ABC, DM \perp BC$ and $DN \perp AB$. Prove that:$DN^2 = DM \times AN$"
- In the given figure, E is a point on side CB produced of an isosceles triangle ABC with $AB = AC$. If $AD \perp BC$ and $EF \perp AC$, prove that $∆ABD \sim ∆ECF$."
- In an isosceles $\triangle ABC$, the base AB is produced both the ways to P and Q such that $AP \times BQ = AC^2$. Prove that $\triangle APC \sim \triangle BCQ$.
- In the given figure, $\frac{QR}{QS}=\frac{QT}{PR}$ and $\angle 1 = \angle 2$. show that $∆PQR \sim ∆TQR$."
- Draw a circle with centre at point $O$. Draw its two chords $AB$ and $CD$ such that $AB$ is not parallel to $CD$. Draw the perpendicular bisectors of $AB$ and $CD$. At what point do they intersect?
Kickstart Your Career
Get certified by completing the course
Get Started