Areas of Parallelograms and Triangles Test

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Areas of Parallelograms and Triangles Test - 14

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Question 1
1 / -0

Use the following information to answer the next question.

In the given figure, BG ⊥ AD and CH ⊥ AD. ABCF and BCDE are parallelogram.

If BG = 12 cm and ED = 6 cm, then what is the area of ΔABF?

A
36 cm²

B
48 cm²

C
60 cm²

D
72 cm²

Solution

Parallelograms on the same base and between the same parallels are equal in area.

∴Area (ABCF) = area (BCDE) = base × height = 6 × 12 cm² = 72 cm²

Area of ΔABF = area (ABCF) = 36 cm²

The correct answer is A.

Question 2

1 / -0

Use the following information to answer the next question.

A parallelogram ABCD and a rectangle ABEF are constructed on the same base AB and the parallel line CF, as shown in the figure. The area of the parallelogram ABCD is A₁ and the area of the rectangle is A₂.

What is the relation between A₁ and A₂?

A₁ = 2A₂

A₂ = 2A₁

A₁ = A₂

A₁ = 3A₂

Solution

A₂ = AB × EB

A₁ = AB × EB

⇒ A₁ = A₂

The correct answer is C.

Question 3
1 / -0

Use the following information to answer the next question.

In the given figure, side RS of parallelogram PQRS is extended on both sides to points T and U, such that triangles PQU and PQT are formed.

If the area of the parallelogram PQRS is 600 cm², then what are the respective areas of ΔPQU and ΔPQT?

A
200 cm² and 200 cm²

B
cm² and cm²

C
300 cm² and 300 cm²

D
cm² and cm²

Solution

In the given figure, it can be seen that ΔPQU and parallelogram PQRS lie on the same base PQ and between the same parallels PQ and UT. Therefore, the area of ΔPQU is equal to half the area of parallelogram PQRS.

∴ Area of ΔPQU = area of parallelogram PQRS cm² = 300 cm²

Similarly, area of ΔPQT = area of parallelogram PQRS = 300 cm²

Thus, the areas of ΔPQU and ΔPQT are 300 cm² and 300 cm² respectively.

The correct answer is C.
Question 4
1 / -0

In the given figure, AB||CD, AC||DE and CE||BD. What is the value of ?

A
2

B
1

C

D

Solution

ACDE is a parallelogram as AC||DE and AE||CD.

Similarly, ECDB is a parallelogram as CE||BD and CD||EB.

Now, parallelograms ACDE and ECDB lie on the same base CD and between the same parallel lines AB and CD.

∴Area of ACDE = Area of ECDB

⇒ Area of ACDE − Area of ΔECD = Area of ECDB − Area of ΔECD

⇒ Area of ΔACE = Area of ΔBDE

Thus, the value of is 1.

Hence, the correct answer is option B.
Question 5
1 / -0

Use the following information to answer the next question.

In the given figure, ABCD is a rectangle and EBCF is a rhombus. The length of CE is 6 cm.

If the area of rectangle ABCD is 42 cm², then what is the measure of BF?

A
7 cm

B
9 cm

C
14 cm

D
18 cm

Solution

It is known that parallelograms on the same base and between the same parallel lines are equal in area.

Now, parallelogram ABCD and EBCF lie on common base BC, and between the same parallel lines BC and AF.

∴ Area of ABCD = Area of EBCF

Thus, the length of BF is 14 cm.

The correct answer is C.
Question 6
1 / -0

Use the following information to answer the next question.

The given figure shows ΔABC and ΔABD that have the same area. It is also given that OC = OD and ∠OBA = 46°.

What is the measure of ∠DAB?

A
58°

B
46°

C
44°

D
38°

Solution

Join CD.

It is seen that ΔABC and ΔABD lie on the same base AB.

It is given that area (ΔABC) = area (ΔABD)

It is known that two triangles lying on the same base and having equal areas lie between the same parallels.

Therefore, AB||CD

∠ABC = ∠BCD (Alternate interior angles)

∴∠BCD = 46°

It is also given that OC = OD.

∴ ∠BCD = ∠ADC (In triangles, angles opposite to equal sides are equal.)

∴∠ADC = 46°

∠DAB = ∠ADC (Alternate interior angles)

∴∠DAB = 46°

Thus, the measure of ∠DAB is 46°.

The correct answer is B.
Question 7
1 / -0

Use the following information to answer the next question.

The given figure shows a parallelogram ABCD where E and F are points on the side AB and DC respectively.

Which of the following relations is true?

A
Area (ΔAFB) = area (ΔDEC)

B
Area (ΔAFB) = 2 area (DEC)

C
2 area (ΔAFB) = 3 area (ΔDEC)

D
3 area (ΔAFB) = 2 area (ΔDEC)

Solution

It is known that if a parallelogram and a triangle lie on the same base and between the same parallels, then the area of the triangle is one-half the area of the parallelogram.

∴Area (ΔAFB) = area (ABCD) ... (1)

Area (ΔDEC) = area (ABCD) ... (2)

From (1) and (2)

Area (ΔAFB) = area (ΔDEC)

The correct answer is A.
Question 8
1 / -0

Use the following information to answer the next question.

In the given figure, PQRS is a quadrilateral. A line is drawn from S parallel to diagonal PR which meets QR produced at T.

What is the value of ?

A
2

B
1

C

D

Solution

It can be seen in the given figure that ΔPRT and ΔPRS are on the same base PR and between the same parallel lines PR and ST.

∴Area of ΔPRT = Area of ΔPRS

⇒ Area of ΔPRT + Area of ΔPQR = Area of ΔPRS + Area of ΔPQR

⇒ Area of ΔPQT = Area of quadrilateral PQRS

The correct answer is B.
Question 9
1 / -0

In the given figure, ABDE and BCDE are parallelograms and EF ⊥ AB and BG ⊥ CD. If AB = 4 cm, EF = 6 cm and CD = 8 cm, then what is the measure of BG?

A
3 cm

B
5 cm

C
6 cm

D
7 cm

Solution

Parallelograms on the same base and between the same parallels are equal in area.

∴Area (ABDE) = area (BCDE)

⇒ AB × EF = BG × CD

⇒ 4 cm × 6 cm = 8 cm × BG

⇒ BG = 3 cm

Hence, the correct answer is option A.
Question 10
1 / -0

Use the following information to answer the next question.

In the given figure, E is any point on the line AB of parallelogram ABCD and AF ⊥ DC.

If AF = 7 cm and AB = 12 cm, then what is the area of ΔDEC?

A
21 cm²

B
28 cm²

C
35 cm²

D
42 cm²

Solution

It is known that if a parallelogram and a triangle lie on the same base and between the same parallels, then the area of the triangle is one-half the area of the parallelogram.

∴Area (ΔDEC) = area (ABCD)

= 42 cm²

The correct answer is D.