RD Sharma solution class 8 chapter 17 Understanding Shapes III(Understanding Special type of Quadrilaterals) Exercise 17.2

Exercise 17.2

Page-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 1:

(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-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 2:

(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 3:

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 4:

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 ∠ACB = 40°, find ∠ADB.

Answer 5:

$$\begin{gathered} \mathrm{In} \mathrm{a} \mathrm{rhombus}, \mathrm{the} \mathrm{diagonals} \mathrm{are} \mathrm{perpendicular}. \\ \therefore \angle \mathrm{BPC}=90° \\ \mathrm{From} ∆ \mathrm{BPC}, \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{angles} \mathrm{is} 180°. \\ \therefore \angle \mathrm{CB}P+\angle \mathrm{BP}C+\angle PBC=180° \\ \angle \mathrm{CB}P=180°-\angle \mathrm{BP}C-\angle PBC \\ \angle \mathrm{CBP}=180°-40°-90°=50° \\ \angle ADB=\angle \mathrm{CBP}=50° (alternate angle) \end{gathered}$$

Question 6:

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

Answer 6:

$$\begin{gathered} \mathrm{All} \mathrm{sides} \mathrm{of} \mathrm{a} \mathrm{rhombus} \mathrm{are} \mathrm{equal} \mathrm{in} \mathrm{length}. \\ \mathrm{The} \mathrm{diagonals} \mathrm{intersect} \mathrm{at} 90° \mathrm{and} \mathrm{the} \mathrm{sides} \mathrm{of} \mathrm{the} \mathrm{rhombus} \mathrm{form} \mathrm{right} \mathrm{triangles}. \\ \mathrm{One} \mathrm{leg} \mathrm{of} \mathrm{these} \mathrm{right} \mathrm{triangles} \mathrm{is} \mathrm{equal} \mathrm{to} 8 \mathrm{cm} \mathrm{and} \mathrm{the} \mathrm{other} \mathrm{is} \mathrm{equal} \mathrm{to} 6 \mathrm{cm}. \\ \mathrm{The} \mathrm{sides} \mathrm{of} \mathrm{the} \mathrm{triangle} \mathrm{form} \mathrm{the} \mathrm{hypotenuse} \mathrm{of} \mathrm{these} \mathrm{right} \mathrm{triangles}. \\ \mathrm{So}, \mathrm{we} \mathrm{get}: \\ (8^2+6^2){\mathrm{cm}}^2 \\ =(64+36){\mathrm{cm}}^2 \\ =100 {\mathrm{cm}}^2 \\ \mathrm{The} \mathrm{hypotneuse} \mathrm{is} \mathrm{the} \mathrm{square} \mathrm{root} \mathrm{of} 100{\mathrm{cm}}^2. \mathrm{This} \mathrm{makes} \mathrm{the} \mathrm{hypotneuse} \mathrm{equal} \mathrm{to} 10. \\ \mathrm{Thus}, \mathrm{the} \mathrm{side} \mathrm{of} \mathrm{the} \mathrm{rhombus} \mathrm{is} \mathrm{equal} \mathrm{to} 10 \mathrm{cm}. \end{gathered}$$

Question 7:

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

Answer 7:

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 40°.

Answer 8:

1. Draw a line segment AB of 3.5 cm.
2. Draw $\angle$BAX equal to 40$°$.
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 9:

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 10:

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 ∆BOC ≅ ∆DOC? State the congruence condition used?
(ii) Also state, if ∠BCO = ∠DCO.

Answer 11:

(i) Yes

$$\begin{gathered} \mathrm{In} ∆\mathrm{BCO} \mathrm{and}∆\mathrm{DCO}: \\ \mathrm{OC}=\mathrm{OC} \left(\mathrm{common}\right) \\ \mathrm{BC}=\mathrm{DC} (\mathrm{all} \mathrm{sides} \mathrm{of} \mathrm{a} \mathrm{rhombus} \mathrm{are} \mathrm{equal}) \\ \mathrm{BO}=\mathrm{OD} (\mathrm{diagonal}s \mathrm{of} \mathrm{a} \mathrm{rhomus} \mathrm{bisect} \mathrm{each} \mathrm{other}) \\ \mathrm{By} \mathrm{SSS} \mathrm{congruence}: \\ ∆\mathrm{BCO} \cong ∆\mathrm{DCO} \end{gathered}$$

(ii) Yes
By c.p.c.t:
$\angle BCO=\angle DCO$

Question 12:

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

Answer 12:

$$\begin{gathered} \mathrm{In} ∆\mathrm{AED} \mathrm{and} ∆\mathrm{DEC}: \\ \mathrm{AE}=\mathrm{EC} (\mathrm{diagonals} \mathrm{bisect} \mathrm{each} \mathrm{other}) \\ \mathrm{AD}=\mathrm{DC} (\mathrm{sides} \mathrm{are} \mathrm{equal}) \\ \mathrm{DE}=\mathrm{DE} \left(\mathrm{common}\right) \\ \mathrm{By} \mathrm{SSS} \mathrm{congruence}: \\ ∆\mathrm{AED}\cong ∆\mathrm{CED} \\ \angle \mathrm{ADE}=\angle \mathrm{CDE} (\mathrm{c}.\mathrm{p}.\mathrm{c}.\mathrm{t}) \\ \mathrm{Similarly}, \mathrm{we} \mathrm{can} \mathrm{prove} ∆\mathrm{AEB} \mathrm{and} ∆\mathrm{BEC}, ∆\mathrm{BEC} \mathrm{and} ∆\mathrm{DEC}, ∆\mathrm{AED} \mathrm{and} ∆\mathrm{AEB} \mathrm{are} \mathrm{congruent} \mathrm{to} \mathrm{each} \mathrm{other}. \\ \mathrm{Hence}, \mathrm{diagonal} \mathrm{of} \mathrm{a} \mathrm{rhombus} \mathrm{bisects} \mathrm{the} \mathrm{angle} \mathrm{through} \mathrm{which} \mathrm{it} \mathrm{passes}. \end{gathered}$$

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 13:

$$\begin{gathered} \mathrm{We} \mathrm{know} \mathrm{that} \mathrm{the} \mathrm{diagonals} \mathrm{of} \mathrm{a} \mathrm{rhombus} \mathrm{bisect} \mathrm{each} \mathrm{other} \mathrm{at} \mathrm{right} \mathrm{angles}. \\ \therefore \mathrm{BO}=\frac{1}{2}\mathrm{BD}=(\frac{1}{2}\times 16) \mathrm{cm} \\ =8\mathrm{cm} \\ \mathrm{AB}=10 \mathrm{cm} \mathrm{and}\angle \mathrm{AOB}=90° \\ \mathrm{From} \mathrm{right} ∆\mathrm{OAB}: \\ {\mathrm{AB}}^2={\mathrm{AO}}^2+{\mathrm{BO}}^2 \\ \Rightarrow {\mathrm{AO}}^2=({\mathrm{AB}}^2–{\mathrm{BO}}^2) \\ \Rightarrow {\mathrm{AO}}^2=(10{)}^2-(8{)}^2{\mathrm{cm}}^2 \\ \Rightarrow {\mathrm{AO}}^2=(100-64) {\mathrm{cm}}^2=36{\mathrm{cm}}^2 \\ \Rightarrow \mathrm{AO}=\sqrt{36} \mathrm{cm}=6\mathrm{cm} \\ \therefore \mathrm{AC}=2\times \mathrm{AO}=(2\times 6) \mathrm{cm}=12 \mathrm{cm} \end{gathered}$$

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 14:

$$\begin{gathered} \mathrm{Let} \mathrm{the} \mathrm{given} \mathrm{quadrilateral} \mathrm{be} \mathrm{ABCD} \mathrm{in} \mathrm{which} \mathrm{diagonals} \mathrm{AC} \mathrm{is} \mathrm{equal} \mathrm{to} 6 \mathrm{cm} \mathrm{and} \mathrm{BD} \mathrm{is} \mathrm{equal} \mathrm{to} 8 \mathrm{cm}. \\ \mathrm{Also}, \mathrm{it} \mathrm{is} \mathrm{given} \mathrm{that} \mathrm{the} \mathrm{diagonals} \mathrm{bisect} \mathrm{each} \mathrm{other} \mathrm{at} \mathrm{right} \mathrm{angle}, \mathrm{at} \mathrm{point} \mathrm{O}. \\ \therefore \mathrm{AO}=\mathrm{OC}=\frac{1}{2}\mathrm{AC}=3 \mathrm{cm} \\ \mathrm{Also}, \mathrm{OB}=\mathrm{OD}=\frac{1}{2}\mathrm{BD}=4 \mathrm{cm} \\ \mathrm{In} \mathrm{right} △\mathrm{AOB}: \\ {\mathrm{AB}}^2={\mathrm{OA}}^2+{\mathrm{OB}}^2 \\ \Rightarrow {\mathrm{AB}}^2=(9+16) {\mathrm{cm}}^2 \\ \Rightarrow {\mathrm{AB}}^2=25 {\mathrm{cm}}^2 \\ \Rightarrow \mathrm{AB}=5 \mathrm{cm} \\ \mathrm{Thus}, \mathrm{the} \mathrm{length} \mathrm{of} \mathrm{each} \mathrm{side} \mathrm{of} \mathrm{the} \mathrm{quadrilateral} \mathrm{is} 5 \mathrm{cm}. \end{gathered}$$