Question 12
Draw a Venn diagram showing the set of parallelograms, kites, rhombuses, rectangles, and squares. Then, answer the following questions—
(i) What is the quadrilateral that is both a kite and a parallelogram? (ii) Can there be a quadrilateral that is both a kite and a rectangle? (iii) Is every kite a rhombus? If not, what is the correct relationship between these two types of quadrilaterals?
We can understand the relationships between different quadrilaterals using a Venn diagram.
Step 1 — Understanding Quadrilateral Relationships and Drawing the Venn Diagram
Let us first define each type of quadrilateral. A parallelogram has two pairs of parallel sides. A kite has two distinct pairs of equal-length adjacent sides. A rhombus is a parallelogram with all four sides equal. A rectangle is a parallelogram with all four angles equal to 90 degrees. A square has all four sides equal and all four angles equal to 90 degrees.
Now, let us find the relationships between these shapes to draw the Venn diagram:
- Every rectangle is a parallelogram.
- Every square is a rectangle.
- Every square is a rhombus.
- Every rhombus is a parallelogram.
- Every rhombus is a kite.
These relationships show us how the sets of these quadrilaterals overlap. The set of squares is a subset of both rectangles and rhombuses. Rectangles and rhombuses are both subsets of parallelograms. Rhombuses are also a subset of kites. This means squares are also a subset of kites.
Step 2 — Answering the Questions
Let us use the Venn diagram and our understanding of quadrilaterals to answer the questions.
Answer
(i) The set of rhombuses is common to both the set of kites and the set of parallelograms. Therefore, a rhombus is both a kite and a parallelogram. (ii) A kite is not a rectangle, and a rectangle is not a kite. Therefore, there can be no quadrilateral that is both a kite and a rectangle. Also, there is no common portion of the set of kites and the set of rectangles. (iii) Every kite is not a rhombus. In the given figure, the kite ABCD is not a rhombus. The correct relationship is that every rhombus is a kite.
More questions in FIO
Find all the other angles inside the following rectangles.
Draw a quadrilateral whose diagonals have equal lengths of 8 cm that bisect each other, and intersect at an angle of
(i) 30° (ii) 40° (iii) 90° (iv) 140°
Consider a circle with centre O. Line segments PL and AM are two perpendicular diameters of the circle. What is the figure APML? Reason and/or experiment to figure this out.
We have seen how to get 90° using paper folding. Now, suppose we do not have any paper but two sticks of equal length, and a thread. How do we make an exact 90° using these?
We saw that one of the properties of a rectangle is that its opposite sides are parallel. Can this be chosen as a definition of a rectangle? In other words, is every quadrilateral that has opposite sides parallel and equal, a rectangle?
Find the remaining angles in the following quadrilaterals.
Using the diagonal properties, construct a parallelogram whose diagonals are of lengths 7 cm and 5 cm, and intersect at an angle of 140°.
Using the diagonal properties, construct a rhombus whose diagonals are of lengths 4 cm and 5 cm.
Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides 4 cm.
Construct a kite whose diagonals are of lengths 6 cm and 8 cm.
Find the remaining angles in the following trapeziums—
Draw a Venn diagram showing the set of parallelograms, kites, rhombuses, rectangles, and squares. Then, answer the following questions—
(i) What is the quadrilateral that is both a kite and a parallelogram? (ii) Can there be a quadrilateral that is both a kite and a rectangle? (iii) Is every kite a rhombus? If not, what is the correct relationship between these two types of quadrilaterals?
If PAIR and RODS are two rectangles, find .
Construct a square with diagonal 6 cm without using a protractor.
CASE is a square. The points U, V, W and X are the midpoints of the sides of the square. What type of quadrilateral is UVWX? Find this by using geometric reasoning, as well as by construction and measurement. Find other ways of constructing a square within a square such that the vertices of the inner square lie on the sides of the outer square, as shown in Figure (b).
If a quadrilateral has four equal sides and one angle of 90°, will it be a square? Find the answer using geometric reasoning as well as by construction and measurement.
What type of a quadrilateral is one in which the opposite sides are equal? Justify your answer.
Hint: Draw a diagonal and check for congruent triangles.
Will the sum of the angles in a quadrilateral such as the following one also be 360°? Find the answer using geometric reasoning as well as by constructing this figure and measuring.
State whether the following statements are true or false. Justify your answers.
(i) A quadrilateral whose diagonals are equal and bisect each other must be a square.
(ii) A quadrilateral having three right angles must be a rectangle.
(iii) A quadrilateral whose diagonals bisect each other must be a parallelogram.
(iv) A quadrilateral whose diagonals are perpendicular to each other must be a rhombus.
(v) A quadrilateral in which the opposite angles are equal must be a parallelogram.
(vi) A quadrilateral in which all the angles are equal is a rectangle.
(vii) Isosceles trapeziums are parallelograms.