A triangle has side lengths of 5 inches, 12 inches, and 15 inches. Every dimension is multiplied by 1/5 to form a new triangle. How is the ratio of the perimeters related to the ratio of corresponding sides?
To determine how the ratio of the perimeters of two triangles is related to the ratio of their corresponding sides, we need to understand the concept of similarity in triangles.
Two triangles are considered similar if their corresponding angles are equal, and the ratios of their corresponding sides are proportional. In other words, if we multiply the dimensions of one triangle by a constant factor, we will obtain a new triangle that is similar to the original triangle.
In this case, you start with a triangle with side lengths of 5 inches, 12 inches, and 15 inches. Then, each dimension is multiplied by 1/5 to form a new triangle. Let's call the side lengths of the new triangle as x, y, and z (in inches).
To find the ratio of the perimeters, compare the sum of the side lengths of the original triangle to the sum of the side lengths of the new triangle.
Original triangle: 5 + 12 + 15 = 32 inches
New triangle: x + y + z
Since every dimension is multiplied by 1/5, we can express the new side lengths in terms of the original side lengths:
x = (1/5) * 5 = 1 inch
y = (1/5) * 12 = 2.4 inches
z = (1/5) * 15 = 3 inches
Now we can find the sum of the side lengths of the new triangle:
New triangle: 1 + 2.4 + 3 = 6.4 inches
Finally, we can compare the ratio of the perimeters:
Ratio of perimeters = (Sum of side lengths of new triangle) / (Sum of side lengths of original triangle)
= 6.4 inches / 32 inches
= 0.2
Therefore, the ratio of the perimeters of the two triangles is 0.2.
In general, the ratio of the perimeters of similar triangles is equal to the ratio of their corresponding sides.