In the figure, $ O $ is the centre of the circle and $ B C D $ is tangent to it at $ C $. Prove that $ \angle B A C+\angle A C D=90^{\circ} $.
"
Given:
In the figure, \( O \) is the centre of the circle and \( B C D \) is tangent to it at \( C \).
To do:
We have to prove that \( \angle B A C+\angle A C D=90^{\circ} \).
Solution:
From the figure,
$\angle ACD = \angle CPA$ (Angles in the alternate segment are equal)
In $\triangle ACP$,
$\angle ACP = 90^o$ (Angle in a semicircle)
$\angle PAC + \angle CPA = 90^o$
$\angle BAC + \angle ACD = 90^o$ ($ACD = ∠CPA$)
Hence proved.
- Related Articles
- \( A B \) is a diameter and \( A C \) is a chord of a circle with centre \( O \) such that \( \angle B A C=30^{\circ} \). The tangent at \( C \) intersects \( A B \) at a point \( D \). Prove that \( B C=B D \).
- In a quadrilateral \( A B C D, \angle B=90^{\circ}, A D^{2}=A B^{2}+B C^{2}+C D^{2}, \) prove that $\angle A C D=90^o$.
- \( A B \) is a chord of a circle with centre \( O, A O C \) is a diameter and \( A T \) is the tangent at \( A \) as shown in the figure. Prove that \( \angle B A T=\angle A C B \)."\n
- In the given figure, \( O C \) and \( O D \) are the angle bisectors of \( \angle B C D \) and \( \angle A D C \) respectively. If \( \angle A=105^{\circ} \), find \( \angle B \)."\n
- Suppose $O$ is the center of a circle and \( A B \) is a diameter of that circle. \( A B C D \) is a cyclic quadrilateral. If \( \angle A B C=65^{\circ}, \angle D A C=40^{\circ} \), then \( \angle B C D=? \)
- In the given figure, $A B C D$ is trapezium with $A B \| D C$. The bisectors of $\angle B$ and $\angle C$ meet at point $O$. Find $\angle B O C$."\n
- In \( \triangle A B C, A D \perp B C \) and \( A D^{2}=B D . C D \).Prove that \( \angle B A C=90^o \)."\n
- In a quadrilateral $A B C D$, $C O$ and $D O$ are the bisectors of $\angle C$ and $\angle D$ respectively. Prove that $\angle C O D=\frac{1}{2}(\angle A+\angle B)$
- In the figure, two tangents \( A B \) and \( A C \) are drawn to a circle with centre \( O \) such that \( \angle B A C=120^{\circ} . \) Prove that \( O A=2 A B \)."\n
- $\triangle A B C$ is an isosceles triangle such that $A B=A C$, $A D \perp B C$a) Prove that $\triangle A B D \cong \triangle A C D$b) Prove that $\angle B=\angle C$c) Is D the mid-point of BC?"\n
- In the figure, \( B C \) is a tangent to the circle with centre \( O \). OE bisects \( A P \). Prove that \( \triangle A E O \sim \triangle A B C \)."\n
- ABCD is a quadrilateral in which \( A D=B C \) and \( \angle D A B=\angle C B A \). Prove that.(i) \( \triangle A B D \cong \triangle B A C \)(ii) \( B D=A C \)(iii) $\angle ABD=\angle BAC$ "\n
- In the figure, the tangent at a point \( C \) of a circle and a diameter \( A B \) when extended intersect at \( P \). If \( \angle P C A=110^{\circ} \), find \( \angle C B A. \)"\n
- In the adjoining figure, \( \Delta A B C \) is an isosceles triangle such that \( A B=A C \). Side BA is produced to D such that $AD=AB$. Show that \( \angle B C D=90^{\circ} \). "\n
- In the figure, $O$ is the centre of the circle, prove that $\angle x = \angle y + \angle z$."\n
Kickstart Your Career
Get certified by completing the course
Get Started
Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies