SOLUTION: If each side of a scalene traingle is tripled then prove that ratio of their area will be 1:9.

Question 1036432: If each side of a scalene traingle is tripled then prove that ratio of their area will be 1:9.
Found 3 solutions by Alan3354, Theo, ikleyn:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
If each side of a scalene traingle is tripled then prove that ratio of their area will be 1:9.
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Use Heron's Law.
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where a, b & c are the sides, and s = (a+b+c)/2
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Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
the ratio of the area is equal to the square of the ratio of the sides.
if the ratio of the sides is 3 to 1, then the ratio of the area is 3^2 to 1 which makes it 9 to 1.

i am assuming that the two scalene triangles are similar.
this means that all of their corresponding sides are proportional.
this includes the height of each of the triangles.

if the larger triangle has sides that are 3 times the size of the smaller triangle, then the height is 3 times the size as well.

the area of any triangle is equal to 1/2 * base * height.

we'll call the base of the smaller triangle b and we'll call the height of the smaller triangle h.

we'll call the base of the larger triangle 3b and we'll call the height of the smaller triangle 3h.

the area of the smaller triangle is 1/2 * b * h
the area of the larger triangle is 1/2 * 3 * b * 3 * h.

simplify the formula and you get the area of the larger triangle is equal to 1/2 * 9 * b * h.

the ratio of the area of the larger triangle is equal to the square of the ratio of the sides of the smaller triangle.

this applies to all triangles, not just scalene triangles.

there is a formula that will calculate the area of a triangle based on the size of the sides alone.

that is heron's formula.

heron's formula says that the area of any triangle can be derived from the length of the sides alone.

the formula states that A = sqrt(s * (s-a) * (s-b) * (s-c))

A is equal to the area of the triangle.
s is equal to (a + b + c) / 2.
a,b,c are the length of the sides of the triangle.

here's a reference to the formula.
https://www.mathsisfun.com/geometry/herons-formula.html
there are many other, more detailed references, that can be found on the web.

if two scalene triangle, or any triangle for that matter, are similar, with their sides in the ratio of 3 to 1, you can use this formula to prove that the areas would be in the ratio of 9 to 1.

here's how i did it.

you are given that heron's formula states that:

A = sqrt(s * (s-a) * (s-b) * (s-c))

A is equal to the area of the triangle.
s is equal to (a + b + c) / 2.

if the two triangle are similar, and their sides are in a ratio of 3 to 1, then every side of the larger triangle is 3 * the size of the smaller triangle.

heron's formula for the larger triangle becomes:

A = sqrt(3*s * (3*s - 3*a) * (3*s - 3*b) * (3*s - 3*c))

this formula can be simplified to:

A = sqrt(3 * s * 3 * (s - a) * 3 * (s - b) * 3 * s-c))

this formula can be further simplified to:

A = sqrt(3 * 3 * 3 * 3 * s * (s - a) * (s - b) * (s - c))

since sqrt(a * b) = sqrt(a) * sqrt(b), this formula can be further simplified to:

A = sqrt(3*3) * sqrt(3*3) * sqrt(s * (s-a) * (s-b) * s-c))

since sqrt(3*3) = sqrt(9) which is equal to 3, then this formula can be further simplified to:

A = 3 * 3 * sqrt(s * (s-a) * (s-b) * s-c)) which can be further simplified to:

A = 9 * sqrt(s * (s-a) * (s-b) * s-c)).

you might also ask where i got 3 * s from.

you are given that s = (a + b + c) / 2

since the larger triangle has sides 3 times the length of the sides of the original triangle, then you get:

s1 = (3 * a + 3 * b + 3 * c) / 2 which can be simplified to:

s1 = 3 * (a + b + c) / 2.

since s = (a + b + c) / 2, it follow that:

s1 = 3 * s because s1 is equal to 3 * (a + b + c) / 2.

since s1 = 3 * s, then that's the reason why 3 * s was used in the formula for the area of the larger triangle.

Answer by ikleyn(52800)   (Show Source): You can put this solution on YOUR website!
.
These are the references to the Heron's formula in this site:
    - Proof of the Heron's formula for the area of a triangle
    - One more proof of the Heron's formula for the area of a triangle

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