Question

(b) \\(\xi = \\{\text{all quadrilaterals}\\}\\)
\\(p = \\{\text{quadrilaterals with at least one pair of parallel sides}\\}\\)
\\(e = \\{\text{quadrilaterals with diagonals that intersect at right angles}\\}\\)

\\(k\\) is a kite.
\\(r\\) is a rhombus.
\\(t\\) is a trapezium.

on the venn diagram below, write \\(k\\), \\(r\\) and \\(t\\) in the appropriate subsets.

Explanation:

Analyze the defined sets

We first identify the geometric properties required for membership in each set:

• Universal set \(\xi\) contains all quadrilaterals.
• Set \(P\) contains quadrilaterals with at least one pair of parallel sides.
• Set \(E\) contains quadrilaterals with diagonals that intersect at right angles (perpendicular diagonals).

Classify the trapezium T

A trapezium is defined as a quadrilateral with at least one pair of parallel sides.

• It belongs to set \(P\) because it has parallel sides.
• Its diagonals do not generally intersect at right angles, so it does not belong to set \(E\).
• Therefore, \(T\) is placed in the region \(P \cap E'\) (inside \(P\) but outside \(E\)).

Classify the kite K

A kite is defined as a quadrilateral with two pairs of adjacent equal sides.

• It does not necessarily have any parallel sides, so it does not belong to set \(P\).
• Its diagonals always intersect at right angles, so it belongs to set \(E\).
• Therefore, \(K\) is placed in the region \(E \cap P'\) (inside \(E\) but outside \(P\)).

Classify the rhombus R

A rhombus is a quadrilateral with four equal sides.

• It is a parallelogram, meaning it has two pairs of parallel sides. Thus, it belongs to set \(P\).
• Its diagonals always intersect at right angles, so it belongs to set \(E\).
• Therefore, \(R\) belongs to both sets and is placed in the intersection region \(P \cap E\).

Answer:

On the Venn diagram, the labels should be placed in the following regions:

• \(T\) (Trapezium) is placed inside circle \(P\) only (in the left region, outside the intersection).
• \(R\) (Rhombus) is placed in the intersection of circle \(P\) and circle \(E\) (the middle overlapping region).
• \(K\) (Kite) is placed inside circle \(E\) only (in the right region, outside the intersection).