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Theorems on Area | Theorems on Area Exercise 14

Question 15

(a) In figure (1) given below, the perimeter of the parallelogram is 42 cm.

Calculate the lengths of the sides of the parallelogram.

(b) In the figure (2) given below, the perimeter of ∆ ABC is 37 cm. If the lengths of

the altitudes AM, BN and CL are 5x, 6x, and 4x respectively, Calculate the lengths

of the sides of ∆ABC.

(c) In the fig. (3) Given below, ABCD is a parallelogram. P is a point on DC such

that area of ∆DAP = 25 cm² and area of ∆BCP = 15 cm². Find

(i) area of || gm ABCD

(ii) DP: PC.

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(a) Given:

The perimeter of parallelogram ABCD = 42 cm

To find:

Lengths of the sides of the parallelogram ABCD.

From fig (1)

We know that,

AB = P

Then, perimeter of ||gm ABCD = 2 (AB + BC)

42 = 2(P + BC)

42/2 = P + BC

21 = P + BC

BC = 21 – P

So, ar (||gm ABCD) = AB × DM

= P × 6

= 6P ………. (1)

Again, ar (||gm ABCD) = BC × DN

= (21 - P) × 8

= 8(21 - P) ………. (2)

From (1) and (2), we get

6P = 8(21 - P)

6P = 168 – 8P

6P + 8P = 168

14P = 168

P = 168/14

= 12

Hence, sides of ||gm are

AB = 12cm and BC = (21 - 12)cm = 9cm

(b) Given:

The perimeter of ∆ ABC is 37 cm. The lengths of the altitudes AM, BN and CL are 5x,

6x, and 4x respectively.

To find:

Lengths of the sides of ∆ABC. i.e., BC, CA, and AB.

Let us consider BC = P and CA = Q

From fig (2),

Then, perimeter of ∆ABC = AB + BC + CA

37 = AB + P + Q

AB = 37 – P – Q

Area (∆ABC) = ½ × base × height

= ½ × BC × AM = ½ × CA × BN = ½ × AB × CL

= ½ × P × 5x = ½ × Q × 6x = ½ (37 – P – Q) × 4x

= 5P/2 = 3Q = 2(37 – P – Q)

Let us consider first two parts:

5P/2 = 3Q

5P = 6Q

5P – 6Q = 0 …… (1)

25P – 30Q (multiplying by 5)….. (2)

Let us consider the second and third parts:

3Q = 2(37 – P – Q)

3Q = 74 – 2P – 2Q

3Q + 2Q + 2P = 74

2P + 5Q = 74 ……. (3)

12P + 30Q = 444 (multiplying by 6)……. (4)

By adding (2) and (4), we get

37P = 444

P = 444/37

= 12

Now, substitute the value of P in equation (1), we get

5P – 6Q = 0

5(12) – 6Q = 0

60 = 6Q

Q = 60/6

= 10

Hence, BC = P = 12cm

CA = Q = 10cm

And AB = 37 – P – Q = 37 – 12 – 10 = 15cm

(c) Given:

ABCD is a parallelogram. P is a point on DC such that area of ∆DAP = 25 cm² and the area

of ∆BCP = 15 cm².

To Find:

(i) area of || gm ABCD

(ii) DP: PC

Now let us find,

From fig (3)

(i) we know that,

ar (∆APB) = ½ ar (||gm ABCD)

Then,

½ ar (||gm ABCD) = ar (∆DAP )

= 25 + 15

\begin{array}{l} =40 \mathrm{cm}^{2} \\ \text { So, ar }(\| \mathrm{gm} \mathrm{ABCD})=2 \times 40=80 \mathrm{cm}^{2} \end{array}

(ii) we know that,

∆ADP and ∆BCP are on the same base CD and between the same parallel lines CD and AB.

ar (∆DAP)/ar(∆BCP) = DP/PC

25/15 = DP/PC

5/3 = DP/PC

So, DP: PC = 5: 3

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Question 15

(a) In figure (1) given below, the perimeter of the parallelogram is 42 cm.

Calculate the lengths of the sides of the parallelogram.

(b) In the figure (2) given below, the perimeter of ∆ ABC is 37 cm. If the lengths of

the altitudes AM, BN and CL are 5x, 6x, and 4x respectively, Calculate the lengths

of the sides of ∆ABC.

(c) In the fig. (3) Given below, ABCD is a parallelogram. P is a point on DC such

that area of ∆DAP = 25 cm² and area of ∆BCP = 15 cm². Find

(i) area of || gm ABCD

(ii) DP: PC.

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(a) Given:

The perimeter of parallelogram ABCD = 42 cm

To find:

Lengths of the sides of the parallelogram ABCD.

From fig (1)

We know that,

AB = P

Then, perimeter of ||gm ABCD = 2 (AB + BC)

42 = 2(P + BC)

42/2 = P + BC

21 = P + BC

BC = 21 – P

So, ar (||gm ABCD) = AB × DM

= P × 6

= 6P ………. (1)

Again, ar (||gm ABCD) = BC × DN

= (21 - P) × 8

= 8(21 - P) ………. (2)

From (1) and (2), we get

6P = 8(21 - P)

6P = 168 – 8P

6P + 8P = 168

14P = 168

P = 168/14

= 12

Hence, sides of ||gm are

AB = 12cm and BC = (21 - 12)cm = 9cm

(b) Given:

The perimeter of ∆ ABC is 37 cm. The lengths of the altitudes AM, BN and CL are 5x,

6x, and 4x respectively.

To find:

Lengths of the sides of ∆ABC. i.e., BC, CA, and AB.

Let us consider BC = P and CA = Q

From fig (2),

Then, perimeter of ∆ABC = AB + BC + CA

37 = AB + P + Q

AB = 37 – P – Q

Area (∆ABC) = ½ × base × height

= ½ × BC × AM = ½ × CA × BN = ½ × AB × CL

= ½ × P × 5x = ½ × Q × 6x = ½ (37 – P – Q) × 4x

= 5P/2 = 3Q = 2(37 – P – Q)

Let us consider first two parts:

5P/2 = 3Q

5P = 6Q

5P – 6Q = 0 …… (1)

25P – 30Q (multiplying by 5)….. (2)

Let us consider the second and third parts:

3Q = 2(37 – P – Q)

3Q = 74 – 2P – 2Q

3Q + 2Q + 2P = 74

2P + 5Q = 74 ……. (3)

12P + 30Q = 444 (multiplying by 6)……. (4)

By adding (2) and (4), we get

37P = 444

P = 444/37

= 12

Now, substitute the value of P in equation (1), we get

5P – 6Q = 0

5(12) – 6Q = 0

60 = 6Q

Q = 60/6

= 10

Hence, BC = P = 12cm

CA = Q = 10cm

And AB = 37 – P – Q = 37 – 12 – 10 = 15cm

(c) Given:

ABCD is a parallelogram. P is a point on DC such that area of ∆DAP = 25 cm² and the area

of ∆BCP = 15 cm².

To Find:

(i) area of || gm ABCD

(ii) DP: PC

Now let us find,

From fig (3)

(i) we know that,

ar (∆APB) = ½ ar (||gm ABCD)

Then,

½ ar (||gm ABCD) = ar (∆DAP )

= 25 + 15

\begin{array}{l} =40 \mathrm{cm}^{2} \\ \text { So, ar }(\| \mathrm{gm} \mathrm{ABCD})=2 \times 40=80 \mathrm{cm}^{2} \end{array}

(ii) we know that,

∆ADP and ∆BCP are on the same base CD and between the same parallel lines CD and AB.

ar (∆DAP)/ar(∆BCP) = DP/PC

25/15 = DP/PC

5/3 = DP/PC

So, DP: PC = 5: 3

Our top 5% students will be awarded a special scholarship to Lido.

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