- Trending Categories
- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP

- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who

# $ \mathrm{ABCD} $ is a rectangle in which diagonal $ \mathrm{AC} $ bisects $ \angle \mathrm{A} $ as well as $ \angle \mathrm{C} $. Show that:

**(i)** $ \mathrm{ABCD} $ is a square

**(ii)** diagonal $ \mathrm{BD} $ bisects $ \angle \mathrm{B} $ as well as $ \angle \mathrm{D} $.

Given:

$ABCD$ is a rectangle in which diagonal $AC$ bisects $\angle A$ as well as $\angle C$.

To do :

We have to show that

(i) $ABCD$ is a square

(ii) Diagonal $BD$ bisects $\angle B$ as well as $\angle D$.

Solution :

(i) A square is a rectangle when all sides are equal.

In the above figure, $AC$ bisects $\angle A$ as well as $\angle C$.

Therefore,

$\angle DAC = \angle BAC$............(i)

$\angle DCA = \angle BCA$............(ii)

In a rectangle, the opposite sides are parallel.

So, $AD \parallel BC$

$AC$ is transversal.

Therefore,

$\angle DAC = \angle BCA$...........(iii) [Alternate interior angles]

From (i) and (iii),

$\angle BCA = \angle BAC$..........(iv)

In $\Delta ABC$,

$\angle BCA = \angle BAC$

So, $AB = BC$...........(v)

We already know that,

In rectangle $ABCD$, $AB = CD, BC = DA$.............(vi)

From (v) and (vi),

$AB = BC = CD = DA$.

Therefore, $ABCD$ is a square.

(ii) $ABCD$ is a square,

Diagonals of a square bisect its angles.

Therefore, Diagonal $BD$ bisects $\angle B$ as well as $\angle D$.

Hence proved.

- Related Questions & Answers
- Biggest Reuleaux Triangle within a Square which is inscribed within a Right angle Triangle?
- Biggest Reuleaux Triangle within a Square which is inscribed within a Right angle Triangle in C?
- Angle between a chord and a tangent when angle in the alternate segment is given in C++?
- Find a time for which angle between hour and minute hands is given theta in C++
- Angle Between Hands of a Clock in C++
- Find d to maximize the number of zeros in array c[] created as c[i] = d*a[i] + b[i] in C++
- Create a data frame with numerical as well as factor columns in R.
- Find d to maximize the number of zeros in array c[] created as c[i] = d*a[i] + b[i] in Python
- Find if there is a rectangle in binary matrix with corners as 1 in C++
- Show that the following grammar is LR (1) S → A a |b A c |B c | b B a A → d B → d
- How to create a vector with lowercase as well as uppercase letters in R?
- Check if it is possible to create a polygon with a given angle in Python
- How do I use `SHOW COLUMNS` as a valid data source for a table?
- Maximum number of squares that can fit in a right angle isosceles triangle in C++
- Count all palindrome which is square of a palindrome in C++
- Area of a circle inscribed in a rectangle which is inscribed in a semicircle in C?