The document RD Sharma Solutions - Chapter 17 - Understanding Shapes-III (Part - 2), Class 8, Maths Class 8 Notes | EduRev is a part of the Class 8 Course RD Sharma Solutions for Class 8 Mathematics.

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**PAGE NO 17.16:**

**Question 1:**

**Which of the following statements are true for a rhombus? (i) It has two pairs of parallel sides. (ii) It has two pairs of equal sides. (iii) It has only two pairs of equal sides. (iv) Two of its angles are at right angles. (v) Its diagonals bisect each other at right angles. (vi) Its diagonals are equal and perpendicular. (vii) It has all its sides of equal lengths. (viii) It is a parallelogram. (ix) It is a quadrilateral. (x) It can be a square. (xi) It is a square.**

**ANSWER:**

(i) True

(ii) True

(iii) True

(iv) False

(v) True

(vi) False

Diagonals of a rhombus are perpendicular, but not equal.

(vii) True

(viii) True

It is a parallelogram because it has two pairs of parallel sides.

(ix) True

It is a quadrilateral because it has four sides.

(x) True

It can be a square if each of the angle is a right angle.

(xi) False

It is not a square because each of the angle is a right angle in a square.

**PAGE NO 17.17:**

**Question 2:**

**Fill in the blanks, in each of the following, so as to make the statement true: (i) A rhombus is a parallelogram in which ...... (ii) A square is a rhombus in which ...... (iii) A rhombus has all its sides of ...... length. (iv) The diagonals of a rhombus ...... each other at ...... angles. (v) If the diagonals of a parallelogram bisect each other at right angles, then it is a ......**

**ANSWER:**

(i) A rhombus is a parallelogram in which __adjacent sides are equal__.

(ii) A square is a rhombus in which __all angles are right angled__.

(iii) A rhombus has all its sides of __equal__ length.

(iv) The diagonals of a rhombus __bisect__ each other at __right__ angles.

(v) If the diagonals of a parallelogram bisect each other at right angles, then it is a __rhombus.__

**Question 3:**

**The diagonals of a parallelogram are not perpendicular. Is it a rhombus? Why or why not?**

**ANSWER:**

No, it is not a rhombus. This is because diagonals of a rhombus must be perpendicular.

**Question 4:**

**The diagonals of a quadrilateral are perpendicular to each other. Is such a quadrilateral always a rhombus? If your answer is 'No', draw a figure to justify your answer.**

**ANSWER:**

No, it is not so.

Diagonals of a rhombus are perpendicular and bisect each other. Along with this, all of its sides are equal. In the figure given below, the diagonals are perpendicular to each other, but do not bisect each other.

**Question 5:**

**ABCD is a rhombus. If âˆ ACB = 40Â°, find âˆ ADB.**

**ANSWER:**

In a rhombus, the diagonals are perpendicular.

âˆ´ âˆ BPC = 90Â°

From Î” BPC, the sum of angles is 180Â°.

âˆ´ âˆ CBP + âˆ BPC + âˆ PBC = 180Â°

âˆ CBP = 180Â°âˆ’âˆ BPCâˆ’âˆ PBC

âˆ CBP = 180Â°âˆ’40Â°âˆ’90Â° = 50Â°

âˆ ADB = âˆ CBP = 50Â° (alternate angle)

**Question 6:**

**If the diagonals of a rhombus are 12 cm and 16cm, find the length of each side.**

**ANSWER:**

All sides of a rhombus are equal in length.The diagonals intersect at 90Â° and the sides of the rhombus form right triangles.

One leg of these right triangles is equal to 8 cm and the other is equal to 6 cm.

The sides of the triangle form the hypotenuse of these right triangles.

So, we get:(82 + 62)cmÂ² = (64 + 36)cmÂ² = 100 cmÂ²

The hypotneuse is the square root of 100cmÂ².

This makes the hypotneuse equal to 10.

Thus, the side of the rhombus is equal to 10 cm.

**Question 7:**

**Construct a rhombus whose diagonals are of length 10 cm and 6 cm.**

**ANSWER:**

1. Draw AC equal to 10 cm.

2. Draw XY, the right bisector of AC, meeting it at O.

3. With O as centre and radius equal to half of the length of the other diagonal,

i.e. 3 cm, cut OB = OD = 3 cm.

4. Join AB, AD and CB, CD.

**Question 8:**

**Draw a rhombus, having each side of length 3.5 cm and one of the angles as 40Â°.**

**ANSWER:**

1. Draw a line segment AB of 3.5 cm.

2. Draw âˆ BAX equal to 40Â°.

3. With A as centre and the radius equal to AB, cut AD at 3.5 cm.

4. With D as centre, cut an arc of radius 3.5 cm.

5. With B as centre, cut an arc of radius 3.5 cm. This arc cuts the arc of step 4 at C.

6. Join DC and BC.

**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.**

**ANSWER:**

1. Draw a line segment AB of 4 cm.

2. Draw a perpendicular XY on AB, which intersects AB at P.

3. With P as centre, cut PE at 3.2 cm.

4. Draw a line WZ that passes through E. This line should be parallel to AB.

5. With A as centre, draw an arc of radius 4 cm that cuts WZ at D.

6. With D as centre and radius 4 cm, cut line DZ. Label it as point C.

4. Join AD and CB.

**Question 10:**

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

**ANSWER:**

1. Draw a line segment AC of 5 cm.

2. With A as centre, draw an arc of radius 6 cm on each side of AC.

3. With C as centre, draw an arc of radius 6 cm on each side of AC. These arcs intersect the arcs of step 2 at B and D.

4. Join AB, AD, CD and CB.

**Question 11:**

**ABCD is a rhombus and its diagonals intersect at O. (i) Is âˆ†BOC â‰… âˆ†DOC? State the congruence condition used? (ii) Also state, if âˆ BCO = âˆ DCO.**

**ANSWER:**

(i) Yes

In Î”BCO and Î”DCO:

OC = OC (common)

BC = DC (all sides of a rhombus are equal)

BO = OD (diagonals of a rhomus bisect each other)

By SSS congruence:Î”BCO â‰… Î”DCO

(ii) Yes

By c.p.c.t:

âˆ BCO = âˆ DCO

**Question 12:**

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

**ANSWER:**

In Î”AED and Î”DEC:

AE = EC (diagonals bisect each other)

AD = DC (sides are equal)

DE = DE (common)

By SSS congruence: Î”AEDâ‰… Î”CED

âˆ ADE = âˆ CDE (c.p.c.t)

Similarly, we can prove Î”AEB and Î”BEC, Î”BEC and Î”DEC, Î”AED and Î”AEB are congruent to each other.

Hence, diagonal of a rhombus bisects 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.**

**ANSWER:**

We know that the diagonals of a rhombus bisect each other at right angles.

âˆ´ BO = 1/2 BD = (1/2Ã—16) cm = 8cm

AB = 10 cm and

âˆ AOB = 90Â°

From right Î”OAB: AB^{2} = AO^{2 }+ BO^{2}

â‡’ AO^{2} = (AB^{2}â€“BO^{2})

â‡’ AO^{2} = (10)^{2}âˆ’(8)^{2}cmÂ²

â‡’ AO^{2} = (100âˆ’64) cmÂ²

= 36cmÂ²

â‡’ AO = cm = 6cm

âˆ´ AC = 2Ã—AO = (2Ã—6) cm = 12 cm

**Question 14:**

**The diagonals 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?**

**ANSWER:**

Let the given quadrilateral be ABCD in which diagonals AC is equal to 6 cm and BD is equal to 8 cm.

Also, it is given that the diagonals bisect each other at right angle, at point O.

âˆ´ AO = OC = 1/2AC = 3 cm

Also, OB = OD = 1/2BD = 4 cm

In right â–³AOB:AB^{2} = OA^{2} + OB^{2}

â‡’ AB^{2 }= (9 + 16) cmÂ²

â‡’ AB^{2} = 25 cmÂ²

â‡’ AB = 5 cmThus, the length of each side of the quadrilateral is 5 cm.