Special Quadrilaterals Geometry

True or False: a rectangle is a type of parallelogram

• true

• false

True or false: a square is a type of rectangle

• true

• false

True or false: a square is a type of quadrilateral

• true

• false

True or false: If a property is true in a rectangle, then it is always true in a parallelogram.

• true

• false

True or false: If a property is true in a rectangle, then it is always true in a square.

• true

• false

FLAT is a rectangle. If FA = 27, find LO.

• 27

• 54

• 13.5

• cannot be determined

True or false: A rhombus is a type of rectangle.

• true

• false

The information given in which of the following statements is sufficient to construct a unique quadrilateral ABCD? Statement I: Quadrilateral ABCD is a square with length of diagonal equal to 5 cm. Statement II: Quadrilateral ABCD is a rhombus with length of diagonal equal to 5 cm.

• Only Statement I

• Only Statement II

• Both Statement I and II together

• Neither statement I nor statement II

Answer explanation

It is possible to construct a square if the measure of a diagonal is given. It is because both the diagonals of a square are equal and perpendicularly bisect each other. This information is sufficient to construct a unique square. In case of rhombus the given information is not sufficient to construct a unique quadrilateral. It is because to construct a unique rhombus minimum two measures are necessary.

In which of the following cases a unique quadrilateral PQRS can be constructed?

• A parallelogram PQRS where PQ = 5.5 cm and QR = 4.2 cm

• A parallelogram PQRS where PQ = 5.5 cm, QR = 4.2 cm and ∠Q = 50°

• A parallelogram PQRS where PR = 6.5 cm, QS = 6.4 cm and ∠QPR = 50°

• All of these

Answer explanation

Using information in option 1, different kind of parallelograms can be produced since there is no limitation on the angle. In option 3, the measures of both diagonals and one angle between diagonal and side is given. But the measures of sides and angles are unknown and can not be determined using properties of parallelogram. Hence, given information is not sufficient to construct a unique parallelogram. The information given in option 2 is sufficient to construct a unique parallelogram. It is because the length of adjacent sides is known and angle between them is also known. Using properties like, opposite sides of a parallelogram are equal and parallel, opposite angles are equal, and the sum of adjacent angles is 180°, we can deduce sufficient information to construct a unique parallelogram. Hence, option is the correct answer.

Construct a rhombus PQRS whose diagonals are PR = 6 cm and QS = 8 cm long. The length of side of the rhombus is ____

• 6 cm

• 5.5 cm

• 4.5 cm

• 5 cm

Answer explanation

Let's construct rhombus PQRS with diagonal PR = 6 cm and diagonal QS = 8 cm. We know that the diagonals of the rhombus are perpendicular bisectors of each other. Using this information we can construct the rhombus PQRS. Step I: draw PR = 6 cm Step II: Draw a line XY which is perpendicular bisector of PR and intersects PR at O. Step III: The length of QS = 8 cm. Since PR bisects QS therefore the length of OS must be 4 cm and OQ is also 4 cm. So, we will find points Q and S on XY such that OS = OQ = 4 cm. For this, take radius of 4cm on the compass and with O as the centre draw an arc which intersects XY at point S. Similarly draw an arc on the opposite side which intersects XY at point Q. Step IV: Join the points P, Q, R and S. using line segments in given order. Thus we have constructed a rhombus PQRS Step V: Measure the length of any one side using compass and ruler. The measure will be 5 cm. Hence, the option 4 is the correct answer.