**Class 9 – Mathematics – Areas of Parallelograms and Triangles**

**IMPORTANT QUESTIONS**

**VERY SHORT ANSWER TYPE QUESTIONS**

**Question.1 Two parallelograms are on equal bases and between the same parallels. Find the ratio of their areas.**

** Solution.** 1:1 [._{.}.Two Parallelograms on the equal based and between the same parallels are equal in area.]

**Question.2 If a triangle and a parallelogram are on same base and between same parallels, then find the ratio of the area of the triangle to the area of parallelogram.**

** Solution.** 1 : 2 [._{.}. If a triangle and a parallelogram are on the same base and between the same parallels, the area of the triangle is equal to half of the parallelogram.]

**Question.3**

**Solution.**

**Question.4.**

**Solution.**

**Question.5**

**Solution.**

**Question.6 In the given figure, D is the mid-point of BC and L mid-is the point of AD. If ar(Î”ABL) = x ar(Î”ABC), then find the value of x**

** **

** Solution.**

**Question.7**

**Solution.**

**Question.8**

**Solution.**

**Question.9**

**Solution.**

**Question.10**

**Solution.**

**Question.11 In figure, TR âŠ¥ PS, PQ // TR and PS // QR. If QR = 8 cm, PQ = 3 cm and SP = 12 cm, find arquad. PQRS).[CBSE-14-17DIG1U]**

** Solution.**

**Question.12**

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**Solution.**

**Question.13**

** Solution.**

**Question.14**

**Solution.**

**SHORT ANSWER QUESTIONS TYPE-I**

**Question.15**

**Solution.**

**Question.16**

**Solution.**

**Question.17 Prove that parallelogram on equal bases and between the same parallels are equal in area. [CBSE March 2012]**

** Solution.**

**Question.18**

**Solution.**

**Question.19**

**Solution.**

**Question.20**

**Solution.**

**Question. 21**

**Solution.**

**Question.22 ABCD is a parallelogram and O is the point of intersection of its diagonals. If ar(Î”AOD) = 4 cm ^{2}, find area of parallelogram ABCD. [CBSE-14-GDQNI3W]**

**Solution.**

**Question.23**

**Solution.**

**Question.24**

**Solution.**

**Question.25**

**Solution.**

**SHORT ANSWER QUESTI-IIONS TYPE**

**Question. 26 A farmer was having a field in the form of a parallelogram PQRS. She took any point A on RS and joined it to points P and Q. In how many parts the field is divided ? What are the shapes of these parts ? The farmer wants to sow wheat and pulses in equal portions of the field separately. How should she do it ?**

** Solution.**

**Question.27**

**Solution.**

**Question.28 InÎ”PQR, A and B are points on side QR such that they trisect QR. Prove that**

** ar(Î”PQB) = 2ar(Î”PBR).**

** Solution.**

**Question.29 For the given figure, check whether the following statement is true or false. Also justify your answer. PQRS is a trapezium with PQ // SR, PS // RU and ST // II RQ, then ar(PURS) = ar(TQRS)**

** [CBSE-14-ERFKZ8H]**

**Solution.**

**Question.30 In the given figure, WXYZ is a quadrilateral with a point P on side WX. If ZY // WX, show that :**

** (i) ar(Î”ZPY) = ar(Î”ZXY)**

** (ii)ar(Î”WZY) = ar(Î”AZPY)**

** (iii)ar(Î”ZWX) = ar(Î”XWY) [CBSE-14-ERFKZ8H]**

**Solution.**

**Question.31. Triangles ABC and DBC are on the same base BC and A, D on the opposite sides of BC, such that ar(Î”ABC) = ar(Î”DBC), show that BC bisect AD.**

** Solution.**

**Question.32 Diagonals AC and BD of a quadrilateral ABCD intersect each other at P. Show that ar(Î”APB) x ar(Î”CPD) = ar(Î”APD) X ar(Î”BPC).**

** Solution.**

**Question.33 The side AB of a parallelogram ABCD is produced to any point P. A line through A and parallel to CP meets CB produced at Q, then parallelogram PBQR is completed (see figure). Show that ar = ar.**

** [CBSE March 2012]**

**Solution.**

**LONG ANSWER TYPE QUESTIONS**

**Question.34. Î”BCD is a parallelogram and P is any point in its interior. Show that ar(Î”APB) + ar(Î”CPD)= ar(Î”BPC) + ar(Î”APD) [CBSE-15-6DWMW5A)**

**Solution.**

**Question.35 In the given figure, ABCD is a square. Side AB is produced to points P and Q in such a way that PA = AB = BQ. Prove that DQ = CP.[CBSE-15-NS72LP7]**

**Solution.**

**Question.36 EFGH is a parallelogram and U and T are points on sides EH and GF respectively. If ar(Î”EHT) = 16 cm ^{2}, find ar(Î”GUF) [CBSE-15-NS72LP7]**

**Solution.**

**Value Based Questions (Solved)**

**Question.1 A farmer having afield in the form of parallelogram PQRS.**

** He planned to built a home for old persons of the village in the field leaving open portion equal to portion covered by the home. For this he divided the field by taking a point A on RS and joining AP, AQ respectively as shown in figure.**

** (i) How should he do it ?**

** (ii)What values are depicted in his plan ?**

**Solution.**

**Question.2 Naveen was having a plot in the shape of a quadrilateral. He decided to donate some portion of it to construct a home for orphan girls. Further he decided to buy a land in lieu of his donated portion of his plot so as to form a triangle.**

** (i) Explain how this proposal will be implemented ?**

** (ii)Which mathematical concept is used in it ?**

** (iii)What values are depicted by Naveen ?**

** Solution.**

(ii) Area of parallelogram and triangle and mid-point theorem.

(iii) Every child, boy or girl have equal right, so avoid discrimination in boy and girl.

**Question.3 Mr Sharma explains his four children two boys and two girls about distribution of his property among them by a picture of A ABC such that D, E, F are mid-points of sides AB, BC, CA respectively are joined to divide A ABC in four triangles as shown in figure.**

** (i) If total property is equal to area of A ABC and share of each child is equal to area of each of four triangles, what does each child has share ?**

** (ii)Which mathematical concept is used in it ?**

** (iii)Which values are depicted in Mr Sharma’s plan ?**

**Solution.**

**Question.4 A flood relief camp was organized by state government for the people affected by the natural calamity near a city. Many school students volunteered to participate in the relief work. In the camp, the food items and first aid centre kits were arranged for the flood victims.**

** The piece of land used for this purpose is shown in the figure.**

** (a) If EFGH is a parallelogram with P and Q as mid-points of sides GH and EF respectively, then show that area used for first aid is half of the total area.**

** (b) What can you say about the student volunteers working for the relief work ?**

** [CBSE-14-GDQNI3W]**

** Solution.**

**Question.5 Sunita has a plot of land which she decides to use for building an old age home and a dispensary for the needy. Her plot is shown in the figure. Plot ABCD is a parallelogram**

** (a) If R is point on diagonals BD. Show that equal areas allotted for building old age home and the dispensary.**

** (b) What value is depicted by the above situation ?**

**Solution.**

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