Class 8 RD Sharma Solutions - Chapter 17 Understanding Shapes Special Types Of Quadrilaterals - Exercise 17.2 | Set 2

In Class 8 mathematics, Chapter 17 of RD Sharma’s textbook explores various special types of quadrilaterals. This chapter is fundamental for the students as it helps in understanding the properties and classifications of the different quadrilaterals which are essential for the solving geometric problems. Exercise 17.2 | Set 2 focuses on applying these properties to solve problems related to the special quadrilaterals.

Understanding Shapes: Special Types of Quadrilaterals

The Special types of quadrilaterals include parallelograms, rectangles, rhombuses, squares, and trapeziums. Each of these quadrilaterals has unique properties:

• Parallelogram: The Opposite sides are equal and parallel.
• Rectangle: All angles are right angles and opposite sides are equal.
• Rhombus: All sides are of equal length and opposite angles are equal.
• Square: All sides and angles are equal it is a special case of both the rectangle and a rhombus.
• Trapezium: Only one pair of the opposite sides is parallel.

These properties are used to solve the various geometric problems including the finding missing angles, side lengths and area calculations.

Question 9. One side of a rhombus is of length 4 cm and the length of an altitude is 3.2 cm. Draw the rhombus.

Solution:

Steps to construct a rhombus:

(i) Draw a line segment of 4 cm

(ii) From point A draw a perpendicular line bisecting the length of 3.2 cm to get point E.

(iii) From point E draw a line parallel to AB.

(iv) From points A and B cut two arcs of length 4 cm on the drawn parallel line to get points D and C.

(v) Join the line segments AD, BC and CD to get rhombus ABCD.

Question 10. Draw a rhombus ABCD, if AB = 6 cm and AC = 5 cm.

Solution:

Steps of construction:

(i) Draw a line segment AB of length 6 cm.

(ii) From point ‘A’ cut an arc of length 5 cm and from point B cut an arc of length 6 cm intersecting at ‘C’.

(iii) Join the line segments AC and BC.

(iv) From point A cut an arc of length 6 cm and from point C cut an arc of 6cm, so that both the arcs intersect at point D.

(v) Join the remaining line segments AD and DC to get rhombus ABCD.

Question 11. ABCD is a rhombus and its diagonals intersect at O.

(i) is ΔBOC ≅ ΔDOC? State the congruence condition used?

(ii) Also state, if ∠BCO = ∠DCO.

Solution:

(i) Yes,

In ΔBOC and ΔDOC

Since, in a rhombus diagonals bisect each other, we have,

BO = DO 

CO = CO Common

BC = CD [All sides of a rhombus are equal]

Now,

By using SSS Congruency, ΔBOC≅ΔDOC

(ii) Yes.

Since by,

∠BCO = ∠DCO, by corresponding parts of congruent triangles.

Question 12. Show that each diagonal of a rhombus bisects the angle through which it passes.

Solution:

(i) In ΔBOC and ΔDOC

BO = DO [In a rhombus diagonals bisect each other]

CO = CO Common

BC = CD [All sides of a rhombus are equal]

By using SSS Congruency, ΔBOC≅ΔDOC

∠BCO = ∠DCO, by corresponding parts of congruent triangles

Therefore, 

Each diagonal of a rhombus bisect the angle through which it passes.

Question 13. ABCD is a rhombus whose diagonals intersect at O. If AB=10 cm, diagonal BD = 16 cm, find the length of diagonal AC.

Solution:

In a rhombus diagonals bisect each other at right angle.

In ΔAOB

BO = BD/2 = 16/2 = 8cm

AB² = AO² + BO² (Pythagoras theorem)

10² = AO² + 8²

100-64 = AO²

AO² = 36

AO = √36 = 6cm

Hence, Length of the diagonal AC is 6 × 2 = 12cm.

Question 14. The diagonal of a quadrilateral are of lengths 6 cm and 8 cm. If the diagonals bisect each other at right angles, what is the length of each side of the quadrilateral?

Solution:

In a rhombus diagonals bisect each other at right angle.

Considering ΔAOB

BO = BD/2 = 6/2 = 3cm

AO = AC/2 = 8/2 = 4cm

Now,

AB² = AO² + BO² (Pythagoras theorem)

AB² = 4² + 3²

AB² = 16 + 9

AB² = 25

AB = √25 = 5cm

Hence, Length of each side of the quadrilateral ABCD is 5cm.

Conclusion

Understanding special types of quadrilaterals is crucial for the solving geometric problems efficiently. Exercise 17.2 | Set 2 in Chapter 17 helps reinforce these concepts by the applying the properties of quadrilaterals to the practical problems. Mastery of these concepts supports the development of the strong problem-solving skills in geometry.