Question

The sides of a triangular field are in the ratio 5: 3: 4 and its perimeter is 180 m. Find:

1. its area.
2. the altitude of the triangle corresponding to its largest side.
3. the cost of leveling the field at the rate of Rs. 10 per square meter.

Answer 1

(i) Given that the sides of a triangle are in the ratio 5: 3: 4.

Also, given that the perimeter of the triangle is 180.

Thus, we have, 5x + 4x + 3x = 180

⇒ 12x = 180

⇒ x = `180/12`

⇒ x = 15.

Thus, the sides are 5 × 15, 3 × 15 and 4 × 15

That is the sides are 75 m, 45 m and 60 m.

Since the sides are in the ratio, 5: 3: 4, it is a Pythagorean triplet,

Therefore, the triangle is a right-angled triangle,

Area of a right-angled triangle = `1/2` × base × altitude

⇒ `1/2`× 45 × 60

⇒ 45 × 30

= 1350 m².

(ii) Consider the following figure.

In the above figure,

The largest side is AC = 75 m,

The altitude corresponding to AC is BD.

We need to find the value of BD.

Now consider the triangles ΔBCD and ΔBAD.
We have,

∠B = ∠B     ...[common]

BD = BD     ...[common]

∠D = ∠D = 90°

Thus, by Angle-Side-Angle criterion of congruence,

we have ΔBCD - ΔABD,

Similar triangles have similar proportionality.

Thus, we have,

`"CD"/"BD" = "BD"/"AD"`

⇒ BD² = AD × CD       ...(1)

From subpart (i), the sides of the triangle are AC = 75 m, AB = 60 m and BC = 45 m

Let AD = x m ⇒ CD = (75 - x) m

Thus applying Pythagoras Theorem,
from right triangle ΔBCD, we have

45² = (75 − x)² + BD²

⇒ BD² = 452 − (75 − x)²

⇒ BD² = 2025 − (5625 + x² − 150x)

⇒ BD² = 2025 − 5625 − x² + 150x

⇒ BD² = −3600 − x² + 150x      ...(2)

Now applying Pythagoras Theorem,
from right triangle ΔABD, we have

60² = x² + BD²

⇒ BD² = 60² - x²

⇒ BD²  = 3600 - x²       ...(3)

From equations (2) and (3), we have,

−3600 − x² + 150x = 3600 − x²

⇒ 150x = 3600 + 3600

⇒ 150x = 7200

⇒ x = `7200/150`

⇒ x = 48

Thus, AD = 48 and CD = 75 − 48 = 27.

Substituting the values AD = 48 m
and CD = 27 m in equation (1), we have

BD² = 48 x 27

⇒ BD² = 1296

⇒ BD = 36 m

The altitude of the triangle corresponding to
its largest side is BD = 36 m

(iii) The area of the triangular field from subpart (i) is 1350 m²

The cost of levelling the field is Rs. 10 per square meter.

Thus the total cost of levelling the field = 1350 × 10 = Rs. 13,500.

Answer 2

a = 5x = 5(15) = 75 m

b = 3x = 3(15) = 45 m

c = 4x = 4(15) = 60 m

P = 180 m

5x + 3x + 4x = 180

12x = 180

x = `180/12 = 15`

S = `180/2` = 90

ar(ABC) = `sqrt("s"("s" - "a")("s" - "b")("s" - "c"))`

`= sqrt(90(90 - 75)(90 - 45)(90- 60))`

`= sqrt(90 xx 15 xx 45 xx 30)`

`= sqrt(9 xx 10 xx 3 xx 5 xx 9 xx 5 xx 3 xx 10)`

`= 9 xx 5 xx 3 xx 10`

= 1350 m²

ar(Δ ABC) = 1350

`1/2 xx "b" xx "h" = 1350`

`1/2 xx 75 xx "h"`= 1350

h = `(1350 xx 2)/75`

h = 18 × 2

h = 36 m

∴ Cost of leveling = 1350 × 10 = 13500