GSEB Class 8 Maths Solutions Chapter 3 Understanding Quadrilaterals Exercise 3.4

Question 1. State whether True or False.
(a) All rectangles are squares.
(b) All rhombuses are parallelograms.
(c) All squares are rhombuses and also rectangles.
(d) All squares are not parallelograms.
(e) All kites are rhombuses.
(f) All rhombuses are kites.
(g) All parallelograms are trapeziums.
(h) All squares are trapeziums.
Answer:
(a) False
(b) True
(c) True
(d) False
(e) False
(f) True
(g) True
(h) True
In simple words: A square always has four equal sides and four right angles. A rectangle only needs four right angles, and a rhombus only needs four equal sides. So, some are special kinds of others, but not all. For example, a square is both a rhombus and a rectangle.

Exam Tip: Remember the specific definitions and properties of each quadrilateral. Visualizing examples can help distinguish between them when determining true or false statements.

Question 2. Identify all the quadrilaterals that have:
(a) Four sides of equal length.
(b) Four right angles.
Answer:
(a) A square and a rhombus both possess four sides that are of equal length.
(b) Squares and rectangles are the quadrilaterals that have four right angles.
In simple words: Only squares and rhombuses have all sides the same length. Only squares and rectangles have all four corner angles as right angles.

Exam Tip: Differentiating quadrilaterals based on side lengths and angle properties is essential. Squares combine both properties, making them unique.

Question 3. Explain how a square is:
1. a quadrilateral
2. a parallelogram
3. a rhombus
4. a rectangle
Answer:
1. A square is a figure with 4 sides, which means it is a quadrilateral.
2. The opposite sides of a square are always equal and also parallel to each other, so it behaves like a parallelogram.
3. All the sides of a square have the same length, so it is considered a rhombus.
4. Every angle in a square is a right angle, so it also functions as a rectangle.
In simple words: A square is like a "super" shape. It has four sides (quadrilateral), opposite sides are parallel (parallelogram), all sides are equal (rhombus), and all angles are 90 degrees (rectangle).

Exam Tip: To fully understand a square, recall the definition of each broader category (quadrilateral, parallelogram, rhombus, rectangle) and show how a square meets all those conditions.

Question 4. Name the quadrilaterals whose diagonals:
1. bisect each other
2. are perpendicular bisectors of each other
3. are equal
Answer:
1. The diagonals of the following quadrilaterals bisect each other: A parallelogram, rectangle, square, and rhombus.
2. The diagonals function as perpendicular bisectors for these quadrilaterals: A square and a rhombus.
3. The diagonals are equal in these cases: A square and a rectangle.
In simple words: Diagonals cut each other in half in parallelograms, rectangles, squares, and rhombuses. They cross at a 90-degree angle AND cut each other in half for squares and rhombuses. They are the same length for squares and rectangles.

Exam Tip: Knowing the unique properties of diagonals for different quadrilaterals is key. Create a mental chart to remember which shape has which diagonal characteristics.

Question 5. Explain why a rectangle is a convex quadrilateral.
Answer:
1. All the angles within a rectangle measure less than 180°.
2. Both of a rectangle's diagonals lie completely within its interior.
Therefore, a rectangle is a convex quadrilateral.
In simple words: A rectangle is convex because all its inside angles are less than 180 degrees. Also, if you draw a line from one corner to the opposite, it always stays inside the shape.

Exam Tip: A convex polygon always has all its interior angles less than 180 degrees, and all its diagonals lie entirely inside the figure. Check for both conditions.

Question 6. ABC is a right-angled 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.)
Answer:
Extend BO to D in such a way that BO = OD. When you join CD and AD, you will get a quadrilateral ABCD where the opposite sides are parallel.
\( \implies \) So, ABCD is a parallelogram.

\( \implies \) We know that \( \angle ABC = 90° \).
\( \implies \) This means ABCD is a rectangle.
Since the diagonals of a rectangle bisect each other, O acts as the mid-point of both BD and AC.
\( \implies \) Therefore, O is equidistant from A, B, and C.
In simple words: If you make the triangle into a rectangle by drawing extra lines, the point O is where the diagonals cross. In a rectangle, the diagonals are equal and cut each other exactly in half. This makes O the same distance from all three original corners (A, B, and C).

Exam Tip: This question uses the property that the midpoint of the hypotenuse of a right triangle is equidistant from all three vertices. Constructing a rectangle helps visualize this geometric principle.