# NCERT Solutions Class 9 Maths Exercise 8.2 of chapter 8- Quadrilateral

**NCERT Solutions Class 9 Maths Exercise 8.2 of chapter 8-Quadrilateral** are the **solutions** of unsolved questions of **exercise 8.2** of the **NCERT maths** text book of **class 9**.All the **solutions** of questions are created by **future study point** by a step by step way for helping the students to boost their preparations for the CBSE board exams.You can also study here **NCERT solutions** of science and **maths** from** class 9** -12, sample papers, **solutions** of previous years question papers, tips for entrance exams of government jobs, carrier in online jobs.

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**NCERT Solutions of class 9 maths**

Chapter 1- Number System | Chapter 9-Areas of parallelogram and triangles |

Chapter 2-Polynomial | Chapter 10-Circles |

Chapter 3- Coordinate Geometry | Chapter 11-Construction |

Chapter 4- Linear equations in two variables | Chapter 12-Heron’s Formula |

Chapter 5- Introduction to Euclid’s Geometry | Chapter 13-Surface Areas and Volumes |

Chapter 6-Lines and Angles | Chapter 14-Statistics |

Chapter 7-Triangles | Chapter 15-Probability |

Chapter 8- Quadrilateral |

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**NCERT Solutions Class 9 Maths Exercise 8.2 of chapter 8- Quadrilateral**

**Q1.ABCD is a quadrilateral in which P, Q, R, and S midpoints of the sides AB, BC, CD, and DA (see the given figure ).AC is a diagonal. Show that:**

(i) SR ll AC, SR = 1/2(AC)

(ii) PQ = SR

(iii)PQRS is a parallelogram

Ans.

**GIVEN: **ABCD is a quadrilateral in which P,Q,R and S mid points of the sides AB,BC, CD and DA

**TO PROVE:**

(i) SR ll AC, SR = 1/2(AC)

(ii) PQ = SR

(iii)PQRS is a parallelogram

**PROOF:**

(i) In ฮADC, R is the mid point point of DC and S is the mid point of AD

According to mid point theorem the line segment joining the mid points of a triangle is parallel to third side and half of the third side in length.

SR ll AC

SR = 1/2(AC)

(ii) In ฮABC ,P is the mid point of AB and Q is the mid point ofย BC ,then using mid point theorem

PQ ll AC

PQ = 1/2(AC)…(i)

SR = 1/2(AC) …(ii)[proved above in (i)]

From equation (i) and (ii)

PQ = SR

(iii) According to mid point theorem,from the figure we have

SR ll AC….(i)

PQ ll AC….(ii)

From equation (i) and (ii) PQ ll SR

PQ = SR (proved above)

According to the theorem of the parallelogram, if one pair of opposite sides of a quadrilateral are equal and parallel, then the other pair of opposite sides is also parallel and equal.

PS ll QR and PS = QR

Therefore PQRS is a parallelogram, Hence proved.

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**Q2.ABCD is a rhombus and P, Q, R, and S are midpoints of the sides AB, BC, CD, and DAย respectively. Show that quadrilateral PQRS is a rectangle.**

Ans.

**GIVEN:** ABCD is a rhombus and P,Q,R and S are midpoints of the sides AB, BC, CD, and DA

**TO PROVE:** PQRS is a rectangle.

**PROOF**: In ฮABD

PSโฅBD (mid point theorem)

โดSU โฅ TO…..(i)

In ฮADC

SRโฅAC (mid point theorem)

โดST โฅ OU….(ii)

From equation (i) and equation (ii),we get that OUST is a parallogram

Therefore โ UST = โ TOU (opposite angle of parallelogram)

โ TOU = 90ยฐ(diagonal of rhombus bisect each other at 90ยฐ)

โดโ UST = 90ยฐ

Similarly โ PQR = โ QRS = โ QPS =90ยฐ

Therefore PQRS is a rectangle, Hence proved

**Q3.ABCD is a rectangle and P,Q,R and S are midpoints of the sides AB,BC,CD and DAย respectively .Show that quadrilateral PQRS is a rhombus.**

Ans.

**GIVEN:**ABCD is a rectangle P,Q,R and S are midpoints of the sides AB,BC,CD and DAย respectively

**TO PROVE:** quadrilateral PQRS is a rhombus

**PROOF:**

In ฮABD, P is the mid point of AB and S is the mid point of AD

Therefore applying mid point theorem

PSโฅBD

In ฮBDC, Q is the mid point of BC and R is the mid point of DC

Therefore applying mid point theorem

QRโฅBD

In ฮADC, S is the mid point of AD and R is the mid point of DC

Therefore applying mid point theorem

SRโฅAC

In ฮABC, P is the mid point of AB and Q is the mid point of BC

Therefore applying mid point theorem

PQโฅAC

AC = BD (diagonal of rectangle)

From equation (i), (ii),(iii) and (iv), we have

PQ = QR = SR = PS

Hence PQRS is a rhombus

**Q4. ABCD is a trapizium in which ABโฅ DC, BD is a diagonal andย E is the mid point of AD. A line is drawn through E parallel to AB intersecting BC at F(see the given figure). Show that F is the mid point of BC.**

Ans.

**GIVEN: **ABโฅ DC

E is the mid point of AD

EFโฅ AB

**TO PROVE:**F is the mid point of BC

**PROOF:** Let EF intersects diagonal BD at G

In ฮABD

EFโฅ AB

โดEG โฅ AB

According to converse of mid point theorem ,If E is the mid point of AD and EG โฅ AB, then G will also the mid point of BD

In ฮBDC

GF โฅ DC (DCโฅ ABโฅEF)

According to converse of mid point theorem ,If G is the mid point of BD and GF โฅ DC, then F will also the mid point of BC

Hence proved

**NCERT Solutions Class 9 Maths Exercise 8.2 of chapter 8- Quadrilateral**

**Q5. In a parallelogram ABCD, E and F are mid point ofย the sides AB and CDย respectively (see the figure). Show that the line segments AF and EC trisect the diagonal BD.ย **

Ans.

**GIVEN:**ABCD is a parallelogram

E is mid point ofย the side AB

F is mid point ofย the side CD

**TO PROVE:** DP = PQ = BQ

**PROOF:** In ABCDย parallelogram

CF = DC/2….(i)(F is mid point ofย the side DC)

AE = AB/2(E is mid point ofย the side AB)

AB = DCย (opposite sides of parallelogram )

AE = DC/2….(ii)

From the equations (i) and (ii)

CF = AE

CF โฅ AE

If a pair of opposite sides of a quadrilateral is equal and parallel then another pair of opposite sides is also equal and parallel

So. in AECF quadrilateral

AF โฅ CE

Therefore in ฮDQC,F is the mid point of DC

PF โฅ CQ ( since AF โฅ CE proved above)

According to converse of mid point theorem ,If F is the mid point of DC and PF โฅ CQ, then P will also the mid point of DQ.

โดDP = PQ….(i)

Similarly ,we can consider ฮAPB and can prove

PQ = BQ…..(ii)

From equation (i) and (ii)

DP = PQ = BQ

Hence proved

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**Q6. Show that line segments joining the mid points of the opposite side of a quadrilateral bisect each other.**

Ans.

**GIVEN:** ABCD is a quadrilateral

P ,Q,R and S are the mid points of the sides AB, BC, DC and AD respectively.

**TO PROVE: **PR and QS bisect each other

**PROOF:**

In ฮABD, P is the mid point of AB and S is the mid point of AD

Therefore applying mid point theorem

PSโฅBD

n ฮBDC, Q is the mid point of BC and R is the mid point of DC

Therefore applying mid point theorem

QRโฅBD

From (i) and (ii)

PS = QR

PS โฅ QR (since PS โฅ BD and QR โฅ BD)

If a pair of opposite sides of a quadrilateral is equal and parallel then another pair of opposite sides is also equal and parallel

Therefore SR = PQ , SR โฅ PQ

Now it is clear that PQRS is a parallelogram

Since diagonal of parallelogram bisect each other

Therefore PR and QS bisect each other, Hence proved

**Q7. ABC is a triangle right angled at C. A line through the mid point M of hypotenuse AB and parallel to BC intersects AC at D. Show that**

**(i) D is the mid point of AC**

**(ii) MD โฅ AC**

Ans.

**GIVEN: ฮ**ABC in which โ C= 90ยฐ

M is mid point of hypotenuse AB

BCโฅ MD

**TO PROVE:**

(i) D is the mid point of AC

According to converse of mid point theorem ,If M is the mid point of AB and BCโฅ MD, then D will also the mid point of AC.

(ii) MD โฅ AC

Since BCโฅ MD (given)

โดโ MDC + โ C = 180ยฐ (sum of co-interior angles)

โ MDC + 90ยฐ= 180ยฐ

โ MDC = 180ยฐ – 90ยฐ = 90ยฐ

Therefore MD โฅ AC

In ฮAMD and ฮCMD ,we have

AD = CD [D is mid point of AC ,proved above in (i)]

DM = DM (common)

โ ADM = โ CDM = 90ยฐ [proved above in (ii)]

ฮADM โ ฮCDM(SAS rule of congruency of triangles)

CM = MA (by CPCT)

Since, M is the mid point of AB

Therefore

Hence proved

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Chapter 1-Sets | Chapter 9-Sequences and Series |

Chapter 2- Relations and functions | Chapter 10- Straight Lines |

Chapter 3- Trigonometry | Chapter 11-Conic Sections |

Chapter 4-Principle of mathematical induction | Chapter 12-Introduction to three Dimensional Geometry |

Chapter 5-Complex numbers | Chapter 13- Limits and Derivatives |

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Chapter 1-Relations and Functions | Chapter 9-Differential Equations |

Chapter 2-Inverse Trigonometric Functions | Chapter 10-Vector Algebra |

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