AJWILL wrote:If the diagnals of a quadrilateral have the same length, is it a square?
1)The diagnals are perpendicular to each other
2)One of the interior angles of the quadrilateral is a right angle.
Please dont just answer the question. i want in depth knowledge on rhombus vs kite vs rectangle vs square to solve variation of these..
Thanks a lot
First of all since you asked, let's review the properties of different figures.
Quadrilateral:
1. A closed polygon of four sides. Sum of interior angles is 360.
Trapeziod/Trapezium:
1. A special quadrilateral in which at least one of the sides is parallel to the other.
2. An isosceles trapezoid has diagonals which are equal in length.
Parallelogram:
1. A special quadrilateral in which opposite sides are parallel and equal.
2. The diagonals may not be equal.
3. Opposite angles are equal.
Rectangle:
1. A special quadrilateral in which opposite sides are equal.
2. All interior angles are equal to 90 degrees.
3. The diagonals are equal and bisect each other.
4. The rectangle is a special case of a parallelogram.
Kite:
1. Adjacent sides are equal
2. Diagonals may not be equal.
3. Diagonals intersect each other at 90 degrees.
4. The longer diagonal bisects the shorter one.
Rhombus:
1. All four sides are equal to each other.
2. Opposite sides are parallel.
3. Diagonals may not be equal.
4. Diagonals bisect each other at 90 degrees.
5. Opposite angles are equal.
Square:
1. All four sides are equal.
2. All internal angles are 90.
3. Diagonals are equal and bisect each other at 90 degrees.
With these in mind, let's look at the question at hand.
We are given that diagonals are equal. We need to find whether the quadrilateral is a square.
1)The diagonals are perpendicular to each other.
The diagonals are equal, and they are perpendicular.It may be a kite or a square. Insufficient.
2)One of the interior angles of the quadrilateral is a right angle.
Since the diagonals are equal, and one the interior angles is 90, it can either be a square or a rectangle. Insufficient.
Together, we know that the diagonals are perpendicular, and one of the interior angles is 90. Now, if we only knew this it could have been a right kite (a kite with one angle = 90, other two obtuse and equal and fourth one less than 90). But we know that longer diagonal of a kite bisects the smaller. Since both diagonals are equal, the bisect each other and hence the probable kite is definitely a square. Sufficient.
Hence
C is the correct answer.
Let me know if this helps
