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ML Aggarwal Solutions Class 9 Mathematics Solutions for Theorems on Area Exercise 14 in Chapter 14 - Theorems on Area
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(a) In the figure (1) given below, ABCD is a parallelogram. Points P and Q on
BC trisect BC into three equal parts. Prove that :
area of ∆APQ = area of ∆DPQ = 1/6 (area of ||gm ABCD)
(b) In the figure (2) given below, DE is drawn parallel to the diagonal AC of the
quadrilateral ABCD to meet BC produced at the point E. Prove that area of quad.
ABCD = area of ∆ABE.
(c) In figure (3) given below, ABCD is a parallelogram. O is any point on the
diagonal AC of the parallelogram. Show that the area of ∆AOB is equal to the area
of ∆AOD.
(a) Given:
From fig (1)
In ||gm ABCD, points P and Q trisect BC into three equal parts.
To prove:
area of ∆APQ = area of ∆DPQ = 1/6 (area of ||gm ABCD)
Firstly, let us construct: through P and Q, draw PR and QS parallel to AB and CD.
Proof:
ar (∆APD) = ar (∆AQD) [Since ∆APD and ∆AQD lie on the same base AD and between
the same parallel lines AD and BC]
ar (∆APD) – ar (∆AOD) = ar (∆AQD) – ar (∆AOD) [On subtracting ar ∆AOD on both
sides]
ar (∆APO) = ar (∆OQD) ….. (1)
ar (∆APO) + ar (∆OPQ) = ar (∆OQD) + ar (∆OPQ) [On adding ar ∆OPQ on both sides]
ar (∆APQ) = ar (∆DPQ) …… (2)
We know that ∆APQ and ||gm PQSR are on the same base PQ and between the same parallel
lines PQ and AD.
ar (∆APQ) = ½ ar (||gm PQRS) …… (3)
Now,
[ar (||gm ABCD)/ar (||gm PQRS)] = [(BC×height)/(PQ×height)] =
[(3PQ×height)/(1PQ×hight)]
ar (||gm PQRS) = 1/3 ar (||gm ABCD) …. (4)
by using (2), (3), (4), we get
ar (∆APQ) = ar (∆DPQ)
= ½ ar (||gm PQRS)
= ½ × 1/3 ar (||gm ABCD)
= 1/6 ar (||gm ABCD)
Hence proved.
(b) Given:
In the figure (2) given below, DE || AC the diagonal of the quadrilateral ABCD to meet at
point E on producing BC. Join AC, AE.ML Aggarwal Solutions for Class 9 Maths
Chapter 14 – Theorems on Area
To prove:
area of quad. ABCD = area of ∆ABE
Proof:
We know that ∆ACE and ∆ADE are on the same base AC and between the same
parallelogram.
ar (∆ACE) = ar (∆ADC)
Now by adding ar (∆ABC) on both sides, we get
ar (∆ACE) + ar (∆ABC) = ar (∆ADC) + ar (∆ABC)
ar (∆ ABE) = ar quad. ABCD
Hence proved.
(c) Given:
From fig (3)
In ||gm ABCD, O is any point on diagonal AC.
To prove:
area of ∆AOB is equal to the area of ∆AOD
Proof:
Let us join BD which meets AC at P.
In ∆ABD, AP is the median.
ar (∆ABP) = ar (∆ADP) ….. (1)
similarly, ar (∆PBO) = ar (∆PDO) …. (2)
now add (1) and (2), we get
ar (∆ABO) = ar (∆ADO) …. (3)
so,
∆AOB = ar ∆AOD
Hence proved.
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