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Practice each question, then open the answer or worked solution.
Diagonals AC and BD of a parallelogram ABCD intersect each other at O. If OA = 3 cm and OD = 2 cm, determine the lengths of AC and BD.
According to the question,
OA = 3 cm
OD = 2 cm
We know that,
Diagonals of parallelogram bisect each other.
Then,
AC = 2AO
AC = 2 × 3 cm
AC = 6 cm
And,
BD = 2OD
BD = 2 × 2 cm
BD = 4 cm
Hence, AC = 6 cm and BD = 4cm
Diagonals of a parallelogram are perpendicular to each other. Is this statement true? Give reason for your answer.
The statement “diagonals of a parallelogram are perpendicular to each other” is false.
Justification:
Diagonals of a parallelogram bisect each other but not at 90°.
So, they are not perpendicular to each other.
Hence, this statement is false.
Can the angles 110º, 80º, 70º and 95º be the angles of a quadrilateral? Why or why not?
The angles 110º, 80º, 70º and 95º cannot be the angles of a quadrilateral.
Justification:
We know that,
Sum of all angles of a quadrilateral = 360°
Sum of given angles,
110° + 80° + 70° + 95° = 355° ≠ 360°
Hence, 110°, 80°, 70° and 95° cannot be the angles of a quadrilateral.
In quadrilateral ABCD, ∠A + ∠D = 180º. What special name can be given to this quadrilateral?
According to the question,
In quadrilateral ABCD, ∠A + ∠D = 180º
We know that,
In a trapezium,
Sum of co-interior angles = 180°
Hence, the given quadrilateral is a trapezium.
All the angles of a quadrilateral are equal. What special name is given to this quadrilateral?
According to the question,
All the angles of a quadrilateral are equal.
Suppose all the angles of the quadrilateral = x
We know that,
Sum of all angles of a quadrilateral = 360°
⇒ x + x + x + x = 360°
⇒ 4x = 360°
⇒ x = 360°/4
⇒ x = 90°
Hence, the quadrilateral is a rectangle.
Diagonals of a rectangle are equal and perpendicular. Is this statement true? Give reason for your answer.
The statement “diagonals of a rectangle are equal and perpendicular” is false.
We know that,
Diagonals of a rectangle bisect each other.
Therefore, they are equal but they are not perpendicular.
Hence, the statement is not true.
Can all the four angles of a quadrilateral be obtuse angles? Give reason for your answer.
All the four angles of a quadrilateral cannot be obtuse angles.
Justification:
We know that,
Sum of all angles of a quadrilateral = 360°
So, at least one angle should be acute angle.
Hence, all the four angles of a quadrilateral cannot be obtuse angles.