The areas of two similar triangles are 50 dm2, and 32 dm2. The sum of their perimeters is 117 dm. What is the perimeter of each of these triangles?

Answer 1

The areas of the two similar figures are in ratio as the square of the scale factor.

√50 / √32

5√2 / 4√2

5 / 4

The perimeters are in the ratio of 5 : 4

5x + 4x = 117

Can you solve for x and answer?

Answer 2

The areas of two similar figures are in ratio as the square of the scale factor.√

√50 / √32 = 5√2 / 4√2 = 5 / 4

The ratio of the perimeters is 5 : 4

5x + 4x = 117

Can you solve for x and answer?

Answer 3

If two objects are similar, then the ratios work like this:

LINEAR ratio is length1 : length2 or height1 : height2 or perimeter1 : perimeter2

If you know the linear ratio, the ratio of the AREAs is the linear ratio SQUARED.

The ratio of the VOLUMEs is the linear ratio CUBED.

Our AREAs are in the ratio 50:32, or 50/32 which reduces to 25/16.

This ratio (because it is the ratio of AREAs) is the SQUARE of the linear ratio.

So if I want the LINEAR ratio, which I do because that will be the ratio of the PERIMETERS, I need to

take the square root of 25/16.

So the LINEAR ratio is 5/4.

So if the smaller triangle has a perimeter of 4, the larger triangle would have a perimeter of 5. But 4 + 5 only = 9, not 117, and the problem states that the sum of their perimeters is 117 dm.

Keeping the same ratio, if the smaller perimeter is 8, the larger would be 10. Sum is 18, but we need 117.

Continuing, if the smaller perimeter is 12, the larger triangle would be 15. Sum is 27, but we need 117.

So we could do this until we reach a sum of 117, but it would be easier with algebra.

Big Perimeter = (5/4) Small Perimeter.

Big Perimeter + Small Perimeter must = 117

Replace the "Big Perimeter" with its equivalent "(5/4) Small Perimeter"

(5/4) Small Perimeter + (1) Small Perimeter = 117

(5/4) Small Perimeter + (4/4) Small Perimeter = 117

(9/4) Small Perimeter = 117

Multiply by reciprocal of (9/4) so that the left side becomes (1) Small Perimeter, and the right side becomes (4/9) 117.

(4/9)(9/4) Small Perimeter = (4/9) 117

(1) Small Perimeter = 52 dm

So Big Perimeter = 117 - 52 = 65 dm

Answer 4

The area of the larger triangle is 50 dm;

(1/2)(Bh) = 50

Bh = 100

The area of the smaller triangle is 32 dm;

Then for some fixed constant k,

32 = (1/2)(Bh)(k^2) = (1/2)(100)(k^2) = 50(k^2)

32/50 = k^2

16/25 = k^2

4/5 = k

So the constant of proportionality is 4/5

The sides of the larger triangle add up to (B+x+y)

The sides of the smaller triangle add up to (B+x+y)*4/5

THe total is (9/5)(B+x+y) = 117

B+x+y = 117/9*5 = 13*5 = 65

The perimeter of the larger triangle is 65.

The perimeter of the smaller triangle is 52