Areas of Parallelograms and Triangles Test

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Areas of Parallelograms and Triangles Test - 1

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Question 1
1 / -0

In the given figure, ar(ABC) = ar(ABD). Which of the following relationships is correct for the lines p and q?

A
p and q are coincident lines.

B
p is parallel to q.

C
p is not parallel to q.

D
p and q are intersecting lines.

Solution

Given: ar(ABC) = ar(ABD) and the triangles are on the same base AB.
Therefore, p || q [Triangles on the same base and having equal area lie between the same parallels]
Question 2
1 / -0

In the given figure, ABCD is a parallelogram. Which of the following relationships is correct if BD is one of its diagonals?

A
ar(ADB) = ar(BCD)

B
ar(ADB) ar(BCD)

C
ar(BCD) = 2ar(ABD)

D
ar(ADB) = 2ar(BCD)

Solution

Given: ABCD is a parallelogram.

AB || CD, AD || BC and BD is a diagonal.
So, ABD and BCD are congruent figures.
Hence, ar(ADB) = ar(BCD) [As two congruent figures have equal areas]
Question 3
1 / -0

In the given figure, A, B, D, C, E and F are points on three lines l, m and n. If ar(ABC) = ar(ADC) = ar(AFC) = ar(AEC), then

A
BD || EF

B
BD ∦ EF

C
AD || CF

D
AD || CE

Solution

Given: ar(ABC) = ar(ADC) = ar(AFC) = ar(AEC)

Since ar(ABC) = ar(ADC) and ABC and ADC are on the same base, BD || AC. …(1)
Since ar(AEC) = ar(AFC) and AEC and AFC are on the same base, AC || EF. … (2)
[As the triangles on the same base and having the same area lie between the same parallels]
From equations (1) and (2), we get BD || EF.
Question 4
1 / -0

In the given figure, line is parallel to DC. ADC and BCD are two triangles on the same base DC. If ADC = BCD = 90°, which of the following relationships is true?

A
ar(ADC) = ar(BCD)

B
ar(ADC) = 2ar(BCD)

C
ar(ADC) ar(BCD)

D
ar(ACD) = ar(ABCD)

Solution

Given: || DC and ADC and BCD are triangles on the same base DC.

Therefore, ar(ADC) = ar(BCD)
[Triangles on the same base and between the same parallels are equal in area]
Question 5
1 / -0

ABCD is a parallelogram. Which of the following relationships is definitely true if P and Q are the midpoints of sides AB and BC, respectively?

A
ar(ABQ) = ar(ADP)

B
ar(APD) = ar(ABCD)

C
ar(ABQ) = 2ar(APD)

D
None of these

Solution

ABCD is a parallelogram. P and Q are the midpoints of AB and BC, respectively.
So, AP = PB = AB and CQ = QB = BC

ar(ADP) = AB x h = ar(ABCD) ... (1)
ar(ABQ) = BC x h₁ = ar(ABCD) ... (2)
From (1) and (2), ar(ADP) = ar(ABQ)
Question 6
1 / -0

PQRS is a quadrilateral and PQ = RS. Which of the following options is correct if ar(APQ) = ar(BPQ)?

A
PQRS is a parallelogram.

B
PQRS is a rhombus.

C
RS is not parallel to PQ.

D
PS is not parallel to RQ.

Solution

PQRS is a quadrilateral in which PQ = RS and ar(APQ) = ar(BPQ).

Since ar(APQ) = ar(BPQ) and the triangles have common base PQ, PQ || RS. [Triangles on the same base and having equal areas lie between the same parallels]
Also, PQ = RS (given)
Thus, PQRS is a parallelogram.
Question 7
1 / -0

In the given figure, AB || CD. Which of the following relationships is definitely true?

A
ar(ΔABC) = 2ar(ΔADB)

B
ar(ΔBCD) = 2ar(ΔABD)

C
ar(ΔABD) = 2ar(ΔAOB)

D
ar(ΔABC) = ar(ΔABD)

Solution

Since AB || CD, and ΔABC and ΔABD have a common base AB.
So, ar(ΔABC) = ar(ΔABD). [Triangles on the same base and lying between the same parallels are equal in area]
Question 8
1 / -0

In the given figure, PQ || SR. If the area of PQR is 24 in², find the area of PQS.

A
12 in²

B
48 in²

C
8 in²

D
24 in²

Solution

PQ || SR, and PQS and PRQ are on the same base PQ, lying between the same parallels.
Therefore, ar(PQS) = ar(PRQ) [Triangles on the same base and lying between the same parallels are equal in area.]
ar(PQS) = 24 in² [ar(PRQ) = 24 in²]
Question 9
1 / -0

In the triangle ABC, E and F are the midpoints of sides AB and AC, respectively.

Which of the following relationships is correct if the area of EOB and the area of FOC are equal?

A
EF ∦ BC

B
EF || BC

C
ar(BOC) = ar(EOF)

D
ar(EOB) = ar(BOC)

Solution

E and F are the respective midpoints of AB and AC of ABC.
Therefore, AE = EB and AF = FC.
Also, ar(EOB) = ar(FOC) (given)
ar(EOB + EOF) = ar(FOC + EOF)
ar(EBF) = ar(CEF)
Also, EBF and CEF lie on the same base EF.
Thus, EF || BC [Triangles on the same base and having equal areas lie between the same parallel lines.]
Question 10
1 / -0

ABC and ABP are on the same base AB. Points C and P are on a line such that AB || . Find ar(ABC) if ar(ABP) = 12 cm².

A
6 cm²

B
12 cm²

C
24 cm²

D
36 cm²

Solution

Given: || AB, ABC and APB are on the same base AB and lie between the same parallels and AB.

So, ar(ABC) = ar(APB) [Triangles on the same base and lying between the same parallels are equal in area.]
ar(ABC) = 12 cm²[∵ar(APB) = 12 cm²]
Question 11
1 / -0

In the given figure, ar(PQR) = ar(MQR). Which of the following relationships is correct?

A
PQ || MR

B
PM || QR

C
PM = QR

D
PQ = MR

Solution

We are given that ar(PQR) = ar(MQR). PQR and MQR lie on the same base QR.
Therefore, PM || QR. [Triangles on the same base and having equal areas lie between the same parallel lines]
Question 12
1 / -0

Two triangles have the same base and equal areas with the third vertex of both of them lying on the same side of the base. Which of the following statements must be true?

A
The two triangles will have another common side.

B
The two triangles will lie between the same parallel lines with the base being coincident with one of them.

C
The two triangles will not lie between the same parallel lines.

D
None of these

Solution

If two triangles have the same base and equal areas, then they must be lying between the same parallel lines.
Question 13
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In the given figure, p || q. Which of the following relationships is true?

A
ar(ADB) = 2ar(ACB)

B
ar(ACB) = 2ar(ADB)

C
ar(ABC) = ar(ABD)

D
ar(AOB) = ar(AOD)

Solution

p || q (Given)
Also, ABC and ABD have a common base AB, and they lie between parallel lines p and q.
Therefore, ar(ABC) = ar(ABD) [If two triangles have the same base and lie between the same parallel lines, then they are equal in area.]
Question 14
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In the given figure, ar(BFC) = ar(BEC). Which of the following relationships is correct?

A
AF ∦ BC

B
AE || BC

C
AE || CD

D
AD || CD

Solution

ar(BFC) = ar(BEC) (Given)

BFC and BEC have a common base BC.
Therefore, AD || BC or AE || BC [If two triangles have the same base and equal areas, then they lie between the same parallel lines.]
Question 15
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In the given figure, || m || n. Which of the following relationships is definitely correct?

A
ar(ADB) ar(ACB)

B
ar(AEG) = ar(ADB)

C
ar(AEF) = ar(GEF)

D
ar(AEG) = ar(ACB)

Solution

|| m || n (Given)
Triangle AEF and Triangle GEF lie between the same parallel lines, m and n, and on the same base EF. So, they are equal in area.
Hence, option (3) is correct.
Question 16
1 / -0

In the given figure, PQ || BC. Which of the following relationships is true?

A
ar(APC) = ar(ABC)

B
2ar(APQ) = ar(ABC)

C
ar(BPQ) = 2ar(CPQ)

D
ar(BPQ) = ar(CPQ)

Solution

PQ || BC (Given)

BPQ and CPQ lie between PQ and BC and have a common base PQ.
Therefore, ar(BPQ) = ar(CPQ) [If two triangles have a common base and they lie between two parallel lines, then areas of the triangles are equal.]
Question 17
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If ABC is a triangle and AD is its median lying on the side BC, then which of the following relationships is correct?

A
ar(ABD) = ar(ABC)

B
ar(ABD) = ar(ADC)

C
ar(ABD) = 2ar(CAB)

D
ar(DAC) = ar(ABC)

Solution

It is given that AD is the median on side BC of ABC.

Therefore, ar(ABD) = ar(ADC) [A median of a triangle divides it into two triangles of equal areas.]
Question 18
1 / -0

PQRS is a parallelogram formed by joining the midpoints of the sides of quadrilateral ABCD. Which of the following relationships is correct?

A
ar(PQRS) = ar(ABCD)

B
ar(PRQ) = ar(PSR)

C
ar(PQR) = ar(DSP)

D
None of the above

Solution

PQRS is a parallelogram formed by joining the midpoints of the sides of quadrilateral ABCD.
Now, parallelogram PQRS and PQR are on the same base QR and lie between the same parallel lines RQ and PS.
So, ar(PRQ) = ar(PQRS) --- (1)
Similarly, for PSR and PQRS:
ar(PSR) = ar(PQRS) --- (2)
[If a parallelogram and a triangle are on the same base and lie between the same parallel lines, then the area of triangle is half the area of parallelogram.]

Comparing (1) and (2), we get
ar(PRQ) = ar(PSR)
Question 19
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If ABCD is a quadrilateral and ar(AOD) = ar(BOC), then which of the following statements is correct?

A
ABCD is a parallelogram.

B
ABCD is a cyclic quadrilateral.

C
ABCD is a trapezium.

D
ABCD is a rhombus.

Solution

ABCD is a quadrilateral.
Also, ar(AOD) = ar(BOC) (Given)
Now, ar(AOD) = ar(BOC)
ar(AOD) + ar(COD) = ar(BOC) + ar(COD)
ar(ADC) = ar(BCD)
Also, ADC and BCD have a common base CD and lie between AB and CD.
Therefore, AB || CD.
[If two triangles are on the same base and have equal areas, then the triangles lie between the same parallel lines.]
Hence, ABCD is a trapezium.
Question 20
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If ar(ADB) : ar(ACB) = 1 : 1, then which of the following relationships is correct?

A
AD || BC

B
AD || AB

C
DC || AB

D
AB || BC

Solution

ar(ADB) : ar(ACB) = 1 : 1 (Given)
ar(ADB) = ar(ACB)
Now, ar(ADB) = ar(ACB), and ABD and ACB have a common base AB.
As they lie between AB and CD, so AB || CD. [If two triangles have a common base and equal areas, then the triangles lie between the same parallel lines.]