## Chapter 8 – Quadrilateral : Constructions and Types

**Practice set 8.1**

**1. Construct the following quadrilaterals of given measures.**

**(1) In □ MORE, l(MO) = 5.8 cm, l(OR) = 4.4 cm, m∠ M = 58°, m∠ O = 105°, m∠ R = 90°.**

**Ans:**

**Rough Figure**

**This is the required construction.**

**(2) Construct □ DEFG such that l(DE) = 4.5 cm, l(EF) = 6.5 cm, l(DG) = 5.5 cm, ****l(DF) = 7.2 cm, l(EG) = 7.8 cm.**

**Ans: **

**Rough Figure**

**This is the required construction.**

**(3) In □ ABCD, l(AB) = 6.4 cm, l(BC) = 4.8 cm, m∠ A = 70°, m∠ B = 50°, m∠ C = 140°.**

**Ans: **

**Rough Figure**

**This is the required construction.**

**(4) Construct □ LMNO such that l(LM) = l(LO) = 6 cm, l(ON) = l(NM) = 4.5 cm, l(OM) = 7.5 cm.**

**Ans: **

**Rough Figure**

**This is the required construction.**

**Practice set 8.2**

**Practice set 8.2**

**1. Draw a rectangle ABCD such that l(AB) = 6.0 cm and l(BC) = 4.5 cm.****Ans:**

**Rough Figure**

**This is the required construction.**

**2. Draw a square WXYZ with side 5.2 cm. ****Ans:**

**Rough Figure**

**This is the required construction.**

**3. Draw a rhombus KLMN such that its side is 4 cm and m∠ K = 75°.****Ans:**

**Rough Figure**

**This is the required construction.**

**4. If diagonal of a rectangle is 26 cm and one side is 24 cm, find the other side.**

**4. If diagonal of a rectangle is 26 cm and one side is 24 cm, find the other side.****Solution:**

**Let □ ABCD be the rectangle.**

**l(BC) = 24cm**

**l(AC) = 26cm**

** **

**In ∆ABC,**

**m∠ABC = 90° … [Angle of a rectangle]**

**∴ By Pythagoras theorem, **

**∴ [l(AC)]² = [l(AB)]2 + [l(BC)]²**

**∴ (26)² = [l(AB)]² + (24)²**

**∴(26)² – (24)² = [l(AB)]²**

**∴[l(AB)]² = 676 – 576**

**∴ [l(AB)]² = 100**

**∴ \(\sqrt{[l(AB)]²}\) = \(\sqrt{100}\) … [Taking square root on both sides]**

**∴ l(AB) = 10 cm**

** **

**Ans:** The length of the other side is 10 cm.

**4. If diagonal of a rectangle is 26 cm and one side is 24 cm, find the other side.**

**4. If diagonal of a rectangle is 26 cm and one side is 24 cm, find the other side.****Solution:**

**Let □ ABCD be the rectangle.**

**l(BC) = 24cm**

**l(AC) = 26cm**

** **

**In ∆ABC,**

**m∠ABC = 90° … [Angle of a rectangle]**

**∴ By Pythagoras theorem **

**∴ [l(AC)]² = [l(AB)]2 + [l(BC)]²**

**∴ (26)² = [l(AB)]² + (24)²**

**∴(26)² – (24)² = [l(AB)]²**

**∴[l(AB)]² = 676 – 576**

**∴ [l(AB)]² = 100**

**∴ \(\sqrt{[l(AB)]²}\) = \(\sqrt{100}\) … [Taking square root on both sides]**

**∴ l(AB) = 10 cm**

** **

**Ans:** The length of the other side is 10 cm.

**5. Lengths of diagonals of a rhombus ABCD are 16 cm and 12 cm. Find the side and perimeter of the rhombus.**

**5. Lengths of diagonals of a rhombus ABCD are 16 cm and 12 cm. Find the side and perimeter of the rhombus.****Solution:**

**In rhombus ABCD,**

**l(AC) = 16 cm**

**l(BD) = 12 cm.**

** **

**Let the diagonals of rhombus ABCD intersect at point O.**

**l(AO) = \(\large \frac {1}{2}\) l(AC) … [Diagonals of a rhombus bisect each other]**

**∴ l(AO) = \(\large \frac {1}{2}\) × 16**

**∴ l(AO) = 8 cm**

** **

**Also, l(DO) = \(\large \frac {1}{2}\) l(BD) … [Diagonals of a rhombus bisect each other]**

**∴ l(DO) = \(\large \frac {1}{2}\) × 12**

**∴ l(DO) = 6 cm**

** **

**In ∆DOA,**

**m∠DOA = 90° .. [Diagonals of a rhombus are perpendicular to each other]**

**∴ By Pythagoras Theorem,**

**[l(AD)]² = [l(AO)]² + [l(DO)]²**

**[l(AD)]² = (8)² + (6)²**

**[l(AD)]² = 64 + 36**

**∴ [l(AD)]² = 100**

**∴ \(\sqrt{[l(AD)]²}\) = \(\sqrt{100}\) … [Taking square root on both sides]**

**∴ l(AD) = 10 cm**

** **

**∴ l(AB) = l(BC) = l(CD) = l(AD) = 10 cm … [Sides of a rhombus are congruent]**

** **

**Perimeter of rhombus ABCD**

**= l(AB) + l(BC) + l(CD) + l(AD)**

**= 10 + 10 + 10 + 10**

**= 40 cm**

** **

**Ans:** The side and perimeter of the rhombus are 10 cm and 40 cm respectively.

**6. Find the length of diagonal of a square with side 8 cm**

**6. Find the length of diagonal of a square with side 8 cm****Solution:**

**Let □ XYWZ is the square of side 8cm.**

**Seg XW is a diagonal.**

** **

**In ∆XYW,**

**m∠XYW = 90° … [Angle of a square]**

**∴ By Pythagoras Theorem,**

**[l(XW)]² = [l(XY)]² + [l(YW)]²**

**∴ [l(XW)]² = (8)² + (8)²**

**∴ [l(XW)]² = 64 + 64**

**∴ [l(XW)]² = 128**

**∴ \(\sqrt{[l(XW)]²}\) = \(\sqrt{128}\) … [Taking square root of both sides]**

**∴ l(XW) = \(\sqrt{64}\) × 2**

**∴ l(XW) = 8 \(\sqrt{2}\) cm**

** **

**Ans:** The length of the diagonal of the square is 8 \(\sqrt{2}\) cm.

**7. Measure of one angle of a rhombus is 50°, find the measures of remaining three angles.**

**7. Measure of one angle of a rhombus is 50°, find the measures of remaining three angles.****Solution:**

**Let □ ABCD be the rhombus.**

**m∠A = 50°**

**m∠C ≅ m∠A … .[Opposite angles of a rhombus are congruent]**

**∴ m∠C = 50°**

**Also, m∠D = m∠B …(i) … .[Opposite angles of a rhombus are congruent]**

**In □ ABCD,**

**m∠A + m∠B + m∠C + m∠D = 360° … [Sum of the measures of the angles of a quadrilateral is 360°]**

**∴ 50° + m∠B + 50° + m∠D = 360°**

**∴ m∠B + m∠D + 100° = 360°**

**∴ m∠B + m∠D = 360° – 100°**

**∴ m∠B + m∠B = 260° … [From (i)]**

**∴ 2m∠B = 260°**

**∴ m∠B = \(\large \frac {260}{2}\)**

**∴ m∠B = 130°**

**∴ m∠D ≅ m∠B = 130° **

**Ans:** The measures of the remaining angles of the rhombus are 130°, 50° and 130°.

**Practice set 8.3**

**Practice set 8.3****1. Measures of opposite angles of a parallelogram are (3x – 2)° and (50 – x)°. Find the measure of its each angle.**

**1. Measures of opposite angles of a parallelogram are (3x – 2)° and (50 – x)°. Find the measure of its each angle.****Solution:**

**Let □ PQRS be the parallelogram.**

**m∠Q = (3x – 2)°**

**m∠S = (50 – x)°**

** **

**m∠Q ≅ m∠S …(i) [Opposite angles of a parallelogram are congruent]**

**∴ 3x – 2 = 50 – x**

**∴ 3x + x = 50 + 2**

**∴ 4x = 52**

**∴ x = \(\large \frac {52}{5}\)**

**∴ x = 13**

** **

**Now, **

**m∠Q = (3x – 2)°**

**∴ m∠Q = (3 × 13 – 2)° **

**∴ m∠Q = (39 – 2)° **

**∴ m∠Q = 37°**

** **

**∴ m∠S ≅ m∠Q = 37° … [From(i)]**

** **

**m∠P + m∠Q = 180°… [Adjacent angles of a parallelogram are supplementary]**

**∴ m∠P + 37° = 180°**

**∴ m∠P = 180° – 37° **

**∴ m∠P = 143°**

** **

**∴ m∠R ≅ m∠P = 143° … [Opposite angles of a parallelogram are congruent]**

** **

**Ans:** The measures of the angles of the parallelogram are 37°, 143°, 37° and 143°.

**2. Referring the adjacent figure of a parallelogram, write the answers of questions given below.**

**2. Referring the adjacent figure of a parallelogram, write the answers of questions given below.**

**(1) If l(WZ) = 4.5 cm then l(XY) = ?**

**(2) If l(YZ) = 8.2 cm then l(XW) = ?**

**(3) If l(OX) = 2.5 cm then l(OZ) = ?**

**(4) If l(WO) = 3.3 cm then l(WY) = ?**

**(5) If m∠WZY = 120° then m∠WXY = ? and m∠ XWZ = ?**

**(5) If m∠WZY = 120° then m∠WXY = ? and m∠ XWZ = ?****Solution:**

**(i) l(WZ) = 4.5 cm … [Given]**

**l(XY) = l(WZ) … [Opposite sides of a parallelogram are congruent ]**

**∴ l(XY) = 4.5 cm**

** **

**(ii) l(YZ) = 8.2 cm … [Given]**

**l(XW) = l(YZ) … [Opposite sides of a parallelogram are congruent]**

**∴ l(XW) = 8.2 cm … [Given]**

** **

**(iii) l(OX) = 2.5 cm … [Given]**

**l(OZ) = l(OX) … [Diagonals of a parallelogram bisect each other]**

**∴ l(OZ) = 2.5 cm**

** **

**(iv) l(WO) = 3.3 cm … [Given]**

**l(WO) = \(\large \frac {1}{2}\) l(WY) … [Diagonals of a parallelogram bisect each other]**

**∴ 3.3 = \(\large \frac {1}{2}\) l(WY)**

**∴ 3.3 × 2 = l(WY)**

**∴ l(WY) = 6.6 cm**

** **

**(v) m∠WZY =120° … [Given]**

**m∠WXY = m∠WZY … [Opposite angles of a parallelogram are congruent]**

**∴ m∠WXY = 120° …(i)**

** **

**m∠XWZ + m∠WXY = 180° … [Adjacent angles of a parallelogram are supplementary]**

**∴ m∠XWZ + 120° = 180° … [From (i)]**

**∴ m∠XWZ = 180°– 120°**

**∴ m∠XWZ = 60°**

**3. Construct a parallelogram ABCD such that l(BC) = 7 cm, m∠ABC = 40°, l(AB) = 3 cm.**

**3. Construct a parallelogram ABCD such that l(BC) = 7 cm, m∠ABC = 40°, l(AB) = 3 cm.****Solution:**

**Opposite sides of a parallelogram are congruent.**

**∴ l(AB) ≅ l(CD) = 3 cm**

**l(BC) ≅ l(AD) = 7 cm**

**Rough Figure**

**This is the required construction.**

**4. Ratio of consecutive angles of a quadrilateral is 1 : 2 : 3 : 4. Find the measure of its each angle. Write, with reason, what type of a quadrilateral it is.**

**4. Ratio of consecutive angles of a quadrilateral is 1 : 2 : 3 : 4. Find the measure of its each angle. Write, with reason, what type of a quadrilateral it is.****Solution:**

**Let □ PQRS be the quadrilateral.**

**Ratio of consecutive angles of a quadrilateral is 1 : 2 : 3 : 4.**

** **

**Let the common multiple be x.**

**∴ m∠P = x°**

**m∠Q = 2x°**

**m∠R = 3x°**

**m∠S = 4x°**

** **

**In □ PQRS,**

**m∠P + m∠Q + m∠R + m∠S = 360° … [Sum of the measures of the angles of a quadrilateral is 360°]**

**∴x + 2x + 3x + 4x = 360**

**∴10x = 360**

**∴ x = \(\large \frac {360}{10}\)**

**∴ x = 36**

** **

**∴ m∠P = x = 36**

**m∠Q = 2x = 2 × 36 = 72**

**m∠R = 3x = 3 × 36 = 108**

**m∠S = 4x = 4 × 36 = 144**

** **

**∴ The measures of the angles of the quadrilateral are 36°, 72°, 108°, 144°.**

** **

**Now,**

**m∠P + m∠S = 36° + 144° = 180°**

** **

**Since interior angles are supplementary,**

**∴ Side PQ || side SR**

**m∠P + m∠Q = 36° + 72° **

**m∠P + m∠Q = 108° ≠ 180°**

** **

**∴ Side PS is not parallel to side QR.**

** **

**Since, one pair of opposite sides of the given quadrilateral is parallel.**

**∴ The given quadrilateral is a trapezium.**

**5. Construct □ BARC such that l(BA) = l(BC) = 4.2 cm, l(AC) = 6.0 cm, l(AR) = l(CR) = 5.6 cm**

**5. Construct □ BARC such that l(BA) = l(BC) = 4.2 cm, l(AC) = 6.0 cm, l(AR) = l(CR) = 5.6 cm****Ans:**

**Rough Figure**

**This is the required construction.**

**6. Construct □ PQRS, such that l(PQ) = 3.5 cm, l(QR) = 5.6 cm, l(RS) = 3.5 cm, m∠ Q = 110°, m∠ R = 70°. If it is given that □ PQRS is a parallelogram, which of the given information is unnecessary?**

**6. Construct □ PQRS, such that l(PQ) = 3.5 cm, l(QR) = 5.6 cm, l(RS) = 3.5 cm, m∠ Q = 110°, m∠ R = 70°. If it is given that □ PQRS is a parallelogram, which of the given information is unnecessary?****Solution:**

**The opposite sides of a parallelogram are congruent.**

**∴ Either l(PQ) or l(SR) is required.**

** **

**To construct a parallelogram, lengths of adjacent sides and measure of one angle is required.**

** **

**∴ Either l(PQ) and m∠Q or l(SR) and m∠R is the unnecessary information given in the question.**

**Rough Figure**

**This is the required construction.**