Understanding Quadrilaterals - 8th math
NCERT Exercise 3.4
Understanding Quadrilaterals NCERT Exercise 3.4 Question (1) State whether True or False.
(a) All rectangles are squares.
Answer False
Explanation A square has all sides equal and all the four angles are 90o, while a rectangle has only opposite side equal and all angles are equal to 90o.
Thus all rectangles are not square and hence the given statement is false.
(b) All rhombuses are parallelograms.
Answer True
Explanation A rhombus has opposite side parallels and equal and diagonals bisect one another. In the case of parallelogram opposite sides are parallel and diagonals bisect one another.
Thus all rhombuses are parallelogram. And hence given statement is true.
(c) All squares are rhombuses and also rectangles.
Answer True
Explanation
A rhombus has opposite side parallels and equal and diagonals bisect one another. And in rectangles opposite sides are parallel and diagonals bisect one another. In a square opposite sides are parallel and diagonals bisect one another.
Thus, all squares are rhombuses and also rectangles. Thus given statement is true.
(d) All squares are not parallelograms.
Answer False
Explanation
In a square opposite sides are parallel and diagonals bisect one another. And in a parallelogram opposite sides are parallel and diagonals bisect one another.
Thus, all squares are parallelogram. And thus given statement is false.
(e) All kites are rhombuses.
Answer False
Explanation
A rhombus has opposite side parallels and equal and diagonals bisect one another. While a quadrilateral with exactly two pair of equal consecutive sides, the diagonals are perpendicular to one another and one of the diagonals bisects the other.
Thus, all kites are not rhombuses. And hence given statement is false.
(f) All rhombuses are kites.
Answer True.
Explanation
A rhombus has opposite side parallels and equal and diagonals bisect one another. While a quadrilateral with exactly two pair of equal consecutive sides, the diagonals are perpendicular to one another and one of the diagonals bisects the other.
Thus, all rhombuses are kite. And hence given statement is true.
(g) All parallelograms are trapezium.
Answer True.
Explanation
A trapezium has at least one pair of opposite sides are parallel. While in a parallelogram opposite sides are parallel.
And hence it can be said that all parallelograms are trapezium but all trapeziums are not parallelograms.
Thus, given statement is true.
(h) All squares are trapeziums.
Answer True.
Explanation
A trapezium has at least one pair of opposite sides are parallel. While in a square opposite sides are parallel.
And hence it can be said that all squares are trapezium but all trapeziums are not squares.
Thus, given statement is true.
Understanding Quadrilaterals NCERT Exercise 3.4 Question (2) Identify all the quadrilaterals that have.
(a) Four sides of equal length.
Answer Square and rhombus.
(b) Four right angles.
Answer Square and rectangle.
Understanding Quadrilaterals NCERT Exercise 3.4 Question (3) Explain how a square is
(i) a quadrilateral.
Answer
A polygon with four sides is called a quadrilateral. Since a square has four sides and thus a square is a quadrilateral.
(ii) a parallelogram
Answer
A parallelogram has opposite sides equal and parallel and diagonals bisect one another. And in a square opposite sides are equal and parallel and diagonals bisect one another.
Thus, a square is a parallelogram.
(iii) a rhombus
A rhombus has all sides are equal, opposite sides are parallel and diagonals bisect one another. And in a square all sides are equal, opposite sides are parallel and diagonals bisect one another.
Thus, a square is a rhombus.
(iv) a rectangle
In a rectangle opposite sides are equal, diagonals are perpendicular bisector. And in a square all sides are equal, this means opposite sides are equal, and diagonals are perpendicular bisector.
Thus, it can be said that a square is a rectangle.
Understanding Quadrilaterals NCERT Exercise 3.4 Question (4) Name the quadrilaterals whose diagonals.
(i) Bisect each other.
Answer A square, a rectangle and a rhombus.
(ii) are perpendicular bisector of each other.
Answer A square, a rectangle and a rhombus.
(iii) are equal
Answer A square and a rectangle.
Understanding Quadrilaterals NCERT Exercise 3.4 Question (5) Explain why a rectangle is a convex quadrilateral.
Answer
In a convex quadrilateral both of the diagonals lies in interior of the quadrilateral. In a rectangle since both of the diagonals lies in the interior, hence a rectangle is a convex quadrilateral.
Understanding Quadrilaterals NCERT Exercise 3.4 Question (6) ABC is a right angle triangle and O is the mid point of the side opposite to the right angle. Explain why O is equidistant from A, B and C. (The dotted lines are drawn additionally to help you).
Solution
Given, ABC is a right angle triangle.
O is the mid-point of the side opposite to right angle.
Thus, to prove O is the equidistant from A, B and C
Or, to prove OA = OB = OB
Construction
Line AB is drawn parallel to BC
And line DC is drawn parallel to AB
BO is extentded to D
Proof
Now, since `bar(AD)\ ||\ bar(BC)`
And, `bar(DC)\ ||\ bar(AB)`
Now since, AD||BC and DC||AB
Thus, &8736;B = ∠D = ∠A = ∠C
Thus, ABCD is a rectangle.
And AC and BD are diagonals of the rectangle ABCD
Thus, both the diagonals AC and BD bisect one another.
Thus, AO = OC = BO = OD
Thus, O lies at equidistance from points A, B and C Proved
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