The lengths of the sides of a triangle are 3, 5, and 7

inches. How many inches long is the shortest side of a
similar triangle that has a perimeter of 75 inches

Then the sides of the similar triangles must be 3x, 5x, and 7x

3x+5x+7x=75
15x=75
x=5

the shortest side is 15

or

perimeter of original = 15
perimeter of new one = 75
so the sides have been increased by a factor of 5
new triangle is 15, 25, and 35

To find the length of the shortest side of a similar triangle, we can use the concept of ratios.

In a similar triangle, the corresponding sides are in proportion. This means that if we scale up or down the original triangle, the ratios of the side lengths will remain the same.

In this case, we have a triangle with side lengths 3, 5, and 7 inches. Let's call the length of the shortest side of the similar triangle as x inches.

To find the ratio, we can take the original triangle's perimeter and divide it by the perimeter of the similar triangle:

Ratio = (Perimeter of Original Triangle) / (Perimeter of Similar Triangle)

Since the original triangle has side lengths 3, 5, and 7 inches and the perimeter is the sum of all the side lengths, the perimeter of the original triangle is 3 + 5 + 7 = 15 inches.

The perimeter of the similar triangle is given as 75 inches.

So, the ratio is: Ratio = 15 / 75 = 1/5

Now, we can set up a proportion using the ratios of the corresponding sides:

Ratio of shortest sides = Ratio of perimeters

Shortest side of original triangle / x = 1/5

To solve for x, we can cross-multiply:

Shortest side of original triangle = (1/5) * x

Since the shortest side of the original triangle is 3 inches, we can substitute this value into the equation:

3 = (1/5) * x

To solve for x, we can multiply both sides of the equation by 5:

5 * 3 = x

x = 15 inches

Therefore, the shortest side of the similar triangle is 15 inches.