A review class for Area of Triangles and Parallelograms designed for SEE, Gr. 10 final exam of Nepal.

1. Area of Parallelograms Virtual Review Class © Bijendra Bir Karmacharya 2020
त्रिभुज र समानान्तर चतुभुुजका क्षेिहरू
Area of Triangles and Parallelograms
Facilitated by : Bijendra Bir Karmacharya
for
Virtual Classes for SEE students
Presented by
Bhaktapur Municipality
2077 Baishakh
गणित (Mathematics) >> ज्यामितत (Geometry) >>

2. Area of Parallelograms Virtual Review Class © Bijendra Bir Karmacharya 2020
Theorem 1 (साध्य १)
एउटै आधार र उही सिानान्तर रेखाहरू विचिा रहेका सिानान्तर
चतुर्ुुजहरूको क्षेत्रफल बराबर हुन्छ ।
Parallelograms standing in same base and lying between same
parallels have equal area.

3. Area of Parallelograms Virtual Review Class © Bijendra Bir Karmacharya 2020
Given: ABCD and ABEF
are parallelograms.
To Prove: A(ABCD) = A(ABEF)
Proof:
S.N. Statement Reasons
1.
2.
3.
In ΔADF and ΔBCE
i) ∠AFD = ∠BEC
ii) FD = EC
iii) ∠ADF = ∠BCE
ΔADF ≡ ΔBCE
A(ABCD) = A(ABED) + A(BCE)
= A(ABED) + A(ADF)
= A(ABEF)
Corresponding ∠s
FE = DC, and DE sub. on both side
Corresponding ∠s
ASA axiom
Whole Part Axiom
From S.No. (2)
Whole Part Axiom
Q.E.D.

4. Area of Parallelograms Virtual Review Class © Bijendra Bir Karmacharya 2020
ALTERNATIVE PROOF
Given: ABCD and ABEF
are parallelograms.
To Prove: A(ABCD) = A(ABEF)
Construction: Draw height EM
Proof:
S.N. Statement Reasons
1.
2.
3.
A(ABCD) = AB × EM
A(ABEF) = AB × EM
A(ABCD) = A(ABEF)
A = b × h
A = b × h
From S.No. (1) and (2)
Q.E.D.

5. Area of Parallelograms Virtual Review Class © Bijendra Bir Karmacharya 2020
Theorem 2 (साध्य २)
एउटै आधार र उही सिानान्तर रेखाहरू विचिा रहेका त्रत्रर्ुज र
सिानान्तर चतुर्ुुज िध्ये त्रत्रर्ुजको क्षेत्रफल सिानान्तर
चतुर्ुुजको क्षेत्रफलको आधा हुन्छ ।
Area of a triangle standing same base and lying between same
parallels with a parallelogram is half of the area of the
parallelogram.

6. Area of Parallelograms Virtual Review Class © Bijendra Bir Karmacharya 2020
Theorem 3 (साध्य ३)
एउटै आधार र उही सिानान्तर रेखाहरू विचिा रहेका
त्रत्रर्ुजहरूको क्षेत्रफल बराबर हुन्छ ।
Triangles standing in same base and lying between same
parallels have equal area.

7. Area of Parallelograms Virtual Review Class © Bijendra Bir Karmacharya 2020
Supporting Corollaries
(सहायक उपसाध्यहरू)
सिानान्तर चतुर्ुुजलाई यसको विकिुले ठिक आधा गछु ।
Diagonal bisects its parallelogram.
त्रत्रर्ुजलाई यसको िध्ध्यकाले ठिक आधा गछु ।
Median bisects its triangle.

8. Area of Parallelograms Virtual Review Class © Bijendra Bir Karmacharya 2020
Very Short Questions – 1 mark
• एउटै आधार र उही सिानान्तर रेखाहरू विचिा रहेका
सिानान्तर चतुर्ुुजहरूविच के सम्बन्ध हुन्छ लेख्नुहोस् । State
the relation between parallelograms standing in same base
and lying between same parallels.
• एउटै आधार र उही सिानान्तर रेखाहरू विचिा रहेका त्रत्रर्ुज र
सिानान्तर चतुर्ुुज विच के सम्बन्ध हुन्छ लेख्नुहोस ् ।
State the relation between a triangle and a parallelogram
standing in same base and lying between same parallels.

9. Area of Parallelograms Virtual Review Class © Bijendra Bir Karmacharya 2020
Very Short Questions – 1 mark
• एउटै आधार र उही सिानान्तर रेखाहरू विचिा रहेका
त्रत्रर्ुजहरू विच के सम्बन्ध हुन्छ लेख्नुहोस ् ।
State the relation between triangles standing in same base
and lying between same parallels.
• ΔABC िा िध्ध्यका AM णखिंचचएको छ । ΔABM र ΔACM विच
के सम्बन्ध हुन्छ लेख्नुहोस ् ।
In ΔABC, median AM is drawn. State the relation between
the triangles ΔABM and ΔACM.

10. Area of Parallelograms Virtual Review Class © Bijendra Bir Karmacharya 2020
Very Short Questions – 1 mark
एउटै आधार AB र उही सिानान्तर रेखाहरू AB//DP विचिा दुइ
सिानान्तर चतुर्ुुजहरू ABCD र ABPQ अिध्थित छन् । तत
सिानान्तर चतुर्ुुज विच के सम्बन्ध हुन्छ लेख्नुहोस ् ।
Two parallelograms ABCD and ABPQ are standing in same base
AB and lying between same parallels AB // DP. State the
relation between the parallelograms ABCD and ABPQ.

11. Area of Parallelograms Virtual Review Class © Bijendra Bir Karmacharya 2020
Short Questions – 2 marks
• त्रत्रर्ुज PQR को र्ुजा QR को िध्यविन्दु M हो । यठद र ΔPQR
को क्षेत्रफल 72 cm² छ र्ने ΔPQM को क्षेत्रफल कतत होला ?
M is the midpoint of the side QR of ΔPQR. If area of ΔPQR is
72 cm², what will be the area of ΔPQM?

12. Area of Parallelograms Virtual Review Class © Bijendra Bir Karmacharya 2020
• ठदइएको चचत्रिा स.च. PQRS र ΔMQR एउटै आधार र उही
सिानान्तर रेखाहरूको विचिा छन् । यठद ΔMQR को क्षेत्रफल
12 cm² र्ए स.च. PQRS को क्षेत्रफल तनकाल्नुहोस ् ।
In the given figure, parallelogram PQRS and ΔMQR are on
the same base and between the same parallel. If the area of
ΔMQR is 12 cm², find out the area of parallelogram PQRS.

13. Area of Parallelograms Virtual Review Class © Bijendra Bir Karmacharya 2020
• ठदइएको चचत्रिा ABCD िगु र MBC एउटा त्रत्रर्ुज हो । यठद
AB = 8 cm छ र्ने ΔMBC को क्षेत्रफल तनकाल्नुहोस् ।
In the adjoining figure, ABCD is a square and MBC is a
triangle. If AB = 8 cm, then find area of ΔMBC.

14. Area of Parallelograms Virtual Review Class © Bijendra Bir Karmacharya 2020
• ठदइएको चचत्रिा ABCD एउटा स.च. हो जसिा BC को
िध्यविन्दु M छ । यठद ΔABM को क्षेत्रफल 15 cm² र्ए स.च.
ABCD को क्षेत्रफल कतत होला ?
In the adjoining figure, ABCD is a parallelogram, where M is
the midpoint of BC. If area of ΔABM is 15 cm², what will be
the area of the parallelogram ABCD ?

15. Area of Parallelograms Virtual Review Class © Bijendra Bir Karmacharya 2020
Long Questions – 4 marks
• Theoretical proofs of Theorems.
• Experimental Verifications from Area
Theorems are avoided in Curriculum.

16. Area of Parallelograms Virtual Review Class © Bijendra Bir Karmacharya 2020
चतुर्ुुज ABCD को शीर्ु विन्दु C बाट विकिु DB सिंग सिानान्तर
हुने गरी णखचचएको रेखा AB लाइ लम््याइएको रेखाको विन्दु E िा
छोएको छ । अब, चतुर्ुुज ABCD को क्षेत्रफल र त्रत्रर्ुज DAE को
क्षेत्रफल बराबर हुन्छ र्तन प्रिाणित गनुुहोस् ।
The line drawn through the vertex C of the quadrilateral ABCD
parallel to the diagonal DB meets AB produced at E. Now, prove
that area of Quad. ABCD is equal to Area of Triangle DAE.

17. Area of Parallelograms Virtual Review Class © Bijendra Bir Karmacharya 2020
ठदइएको त्रत्रर्ुज ABC िा BE र CD दुइ िध्ध्यकाहरू हुन्, जुन विन्दु
O िा प्रततच्छेठदत छन् । अब, त्रत्रर्ुज BOC र चतुर्ुुज ADOE का
क्षेत्रफलहरू बराबर छन् र्तन मसद्ध गनुुहोस् ।
In the given triangle ABC, two medians BE and CD are
intersecting at O. Proved that the areas of triangle BOC and
quadrilateral ADOE are equal.

18. Area of Parallelograms Virtual Review Class © Bijendra Bir Karmacharya 2020
धन्यिाद
विजेन्रिीर किाुचायु
श्री चक्रे श्िर िा.वि., बेिानचोक-१, काभ्रे
9841271567
veejayandra@gmail.com
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