The sides of a triangle are in the ratio of 3:4:5 if its perimeter is 72 cm then what is its area

This discussion on The sides of a triangle are in the ratio of 3 : 4 : 5. If its perimeter is 36 cm, then what is its area?a)32 cm2b)54 cm2c)67 cm2d)72cm2Correct answer is 'B'. Can you explain this answer? is done on EduRev Study Group by Class 9 Students. The Questions and Answers of The sides of a triangle are in the ratio of 3 : 4 : 5. If its perimeter is 36 cm, then what is its area?a)32 cm2b)54 cm2c)67 cm2d)72cm2Correct answer is 'B'. Can you explain this answer? are solved by group of students and teacher of Class 9, which is also the largest student community of Class 9. If the answer is not available please wait for a while and a community member will probably answer this soon. You can study other questions, MCQs, videos and tests for Class 9 on EduRev and even discuss your questions like The sides of a triangle are in the ratio of 3 : 4 : 5. If its perimeter is 36 cm, then what is its area?a)32 cm2b)54 cm2c)67 cm2d)72cm2Correct answer is 'B'. Can you explain this answer? over here on EduRev! Apart from being the largest Class 9 community, EduRev has the largest solved Question bank for Class 9.

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Given,

Sides ratio of a triangle = 3 : 4 : 5

Area of the triangle = 216 cm2

Formula:

Area of triangle = √[s (s - a) (s - b) (s - c)]

Semi-perimeter = (a + b + c)/2

Perimeter of the triangle = a + b + c

Calculation:

Sides ratio of triangle = 3x : 4x : 5x

⇒ s = (3x + 4x + 5x)/2

⇒ s = 12x/2 = 6x

Area of triangle = √ [6x × (6x - 3x) × (6x - 4x) × (6x - 5x)]

⇒ 216 = √ [6x × 3x × 2x × x]

⇒ 216 = 6x2

⇒ 6x2 = 216

⇒ x2 = 216/6

⇒ x = √36

⇒ x = 6

∴ Perimeter of the triangle = 3x + 4x + 5x = 12x = 12 × 6 = 72 cm

Smart method

As we know,  3 : 4 : 5 are triplets. Hence, let sides be 3x, 4x and 5x.

Now, area of triangle = 1/2 × 3x × 4x = 216

⇒ 6x2 = 216

⇒ x2 = 36

⇒ x = 6

∴ Perimeter of the triangle = 3x + 4x + 5x = 12x = 12 × 6 = 72 cm

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Perimeter P of the triangle = P = 72cmRatio of the sides = 3:4:5Let the constant of proportionality be k⇒ 3k + 4k + 5k = 72

⇒ 12k = 72

⇒ k = `(72)/(12)`= 6∴ the sides are: 3 x 6, 4 x 6 and 5 x 6I.e. 18cm, 24cm and 30cm

We know that, Area of a Triangle whose sides are a, b, and c and semiperimeter is s  given by `sqrt("s"("s" - "a")("s" - "b")("s" - "c")); "s" = ("a" + "b" + "c")/(2)`

For a triangle whose sides are 18cm, 24cm and 30cm

i.e. a = 18 b = 24 and c = 30, s = `(72)/(2)` = 36

Area= `sqrt(36(36 - 18)(36 - 24)(36 - 30)`= `sqrt(36(18)(12)(6)`

= 21.6cm2 

We also know that, Area of a Triangle = `(1)/(2)"b.h"  "i.e." (1)/(2)("Base" xx "Height")`

Area of a Traingle with base = 30cm and height = h cm

⇒ `(1)/(2)15."h"` = 216cm2

⇒ h = `(216 xx 2)/(30)`
= 14.4cm.

We know the perimiter is split into the proportions 3:4:5 therefore by adding 3+4+5 we can work out what fraction of 72cm each side is.Thus 3/(3+4+5) = 3/12 = 1/4 is the fraction of 72 that belongs to the shortest side. This means the shorest side has lenght of 1/4 * 72 = 18cm.We can repeat this to work out the lenght of the other two sides giving the middle sized side to have lenght 4/12 * 72 = 1/3 * 72 = 24cm.And the longest side lenght 5/12 * 72 = 30cm.
We can now draw and label a sketch of the triangle.As it is a right anlged triangle we can use the formular Area = 1/2 * Base * Height. Which gives 24 * 18 * 1/2 = 12 * 18 = 10 * 18 + 2 * 18 = 216cm2