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
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Solution.
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Question.4.
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Solution.
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Question.5
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Solution.
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Question.6 In the given figure, D is the mid-point of BC and L mid-is the point of AD.  If ar(ΔABL) = x ar(ΔABC), then find the value of x
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Solution.
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Question.7
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Solution.
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Question.8
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Solution.
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Question.9
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Solution.
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Question.10
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Solution.
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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.
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Question.12
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Solution.
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Question.13
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Solution.
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Question.14
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Solution.
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SHORT ANSWER QUESTIONS TYPE-I

Question.15
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Solution.
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Question.16
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Solution.
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Question.17 Prove that parallelogram on equal bases and between the same parallels are equal  in area. [CBSE March 2012]
Solution.
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Question.18
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Solution.
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Question.19
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Solution.
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Question.20
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Solution.
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Question. 21
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Solution.
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Question.22 ABCD is a parallelogram and O is the point of intersection of its diagonals. If ar(ΔAOD) = 4 cm2, find area of parallelogram ABCD. [CBSE-14-GDQNI3W]
Solution.
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Question.23
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Solution.
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Question.24
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Solution.
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Question.25
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Solution.
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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.
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Question.27
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Solution.
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Question.28 InΔPQR, A and B are points on side QR such that they trisect QR. Prove that
ar(ΔPQB) = 2ar(ΔPBR).
Solution.
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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]
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Solution.
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Question.30 In the given figure, WXYZ is a quadrilateral with a point P on side WX. If ZY // WX, show that :
(i) ar(ΔZPY) = ar(ΔZXY)
(ii)ar(ΔWZY) = ar(ΔAZPY)
(iii)ar(ΔZWX) = ar(ΔXWY)  [CBSE-14-ERFKZ8H]
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Solution.
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Question.31. Triangles ABC and DBC are on the same base BC and A, D on the opposite sides of BC, such that ar(ΔABC) = ar(ΔDBC), show that BC bisect AD.
Solution.
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Question.32 Diagonals AC and BD of a quadrilateral ABCD intersect each other at P. Show that ar(ΔAPB) x ar(ΔCPD) = ar(ΔAPD) X ar(ΔBPC).
Solution.
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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({ \parallel  }^{ gm }ABCD) = ar({ \parallel  }^{ gm }ABCD) .
[CBSE March 2012]
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Solution.
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LONG ANSWER TYPE QUESTIONS

Question.34. ΔBCD is a parallelogram and P is any point in its interior. Show that ar(ΔAPB) + ar(ΔCPD)= ar(ΔBPC) + ar(ΔAPD) [CBSE-15-6DWMW5A)
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Solution.
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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]
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Solution.
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Question.36 EFGH is a parallelogram and U and T are points on sides EH and GF respectively. If ar(ΔEHT) = 16 cm2, find ar(ΔGUF) [CBSE-15-NS72LP7]
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Solution.
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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 ?
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Solution.
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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.
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(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 ?
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Solution.
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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]
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Solution.
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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 ?
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Solution.
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