The Perimeter of a triangle is 540metre and its sides are in the ratio 12 25 17 find the are of the triangle?

Answer 1

What does the ratio 12 25 17 mean? It simply means that after dividing the original three sides by the same unknown number, we get the values of 12, 25 and 17.

Lets call this unknown number x:

(Side a) / x = 12. So Side a = 12x( multiply both sides by unknown x)

(Side b) / x = 25. So Side b = 25x( multiply both sides by unknown x)

(Side c)/ x =17. So Side c = 17x( multiply both sides by unknown x)

We also know that the perimeter of the triangle (Side a + Side b + Side c) = 540.

We know each side as a multiple of x, so: 12x + 25x + 17x = 540.

54x = 540.

x =10.

Now that we know the number each side was divided by to get the values 12 25 and 17, we can now get the original lengths.

Side a =12x = 12 multiplied by 10 = 120meter.

Side b =25x = 25 multiplied by 10 = 250meter.

Side c =17x = 17 multiplied by 10 = 170meter.

To double check your answer, add up all three sides and you get the perimeter of 540meters.

Now you need to find the area of the triangle ABC with sides a = 120 m, b = 250 m, and c = 170 m.

From the vertex B, draw the perpendicular BD (the height h) to the base b. Two right triangles with hypotenuses c and a are formed.

Use the Law of Cosine to find the measure of the angle A, such that

cos A = (b2 + c2 - a2)/2bc = (2502 + 1702 - 1202)/2(250)(170) = 77,000/85,000

cos A ≈ 0.906

A = cos-1 0.906 ≈ 25⁰

Look at the formed right triangle ABD, here we have

sin A = h/c which yields

h = c sin A (substitute the known information)

h = 170 sin 25⁰ ≈ 72 m

Now you are able to find the area of the triangle ABC.

A = bh/2 = (250 m)(72 m)/2 = 9,000 m2