A triangle has the side lengths of 5 inches 12 inches and 15 inches every demention is multiplyed by 1/5 to form a new triangle how is the ratio of the perimeters related to the ratios corresponding sides?
Well, if you're thinking of shrinking a triangle by multiplying each dimension by 1/5, it's like giving it a little "triangle diet". It's going on a scaling down operation! Now, let me put on my clown hat and explain the relationship between the ratios of the perimeters and the corresponding sides.
When you scale down the dimensions of a triangle by a factor of 1/5, you're essentially shrinking it like a pair of pants in the wash. In this case, the sides of the original triangle are 5 inches, 12 inches, and 15 inches. So, if we multiply each of these sides by 1/5, we get a new triangle with sides measuring 1 inch, 2.4 inches, and 3 inches.
Now, let's talk about the ratios. The ratio of the perimeters of the original triangle and the scaled down triangle is the same as the ratio of the corresponding sides. In this case, the scaled down triangle sides are 1 inch, 2.4 inches, and 3 inches, while the original triangle sides were 5 inches, 12 inches, and 15 inches.
So, if we compare the corresponding sides, we have:
1 inch : 5 inches
2.4 inches : 12 inches
3 inches : 15 inches
And if we simplify these ratios, we get:
1 : 5
2.4 : 12 (which can be further simplified to 1 : 5)
3 : 15 (which can also be simplified to 1 : 5)
Ta-da! As you can see, the ratios of the perimeters and the ratios of the corresponding sides of the triangles are the same. So, whether you're talking about the perimeters or the sides, they all maintain that magical ratio of 1 : 5.
To understand how the ratio of the perimeters is related to the ratio of the corresponding sides, we need to calculate the perimeters of the two triangles.
Let's denote the original triangle as Triangle ABC, with side lengths AB = 5 inches, BC = 12 inches, and AC = 15 inches.
If we multiply each dimension by 1/5, we get a new triangle, Triangle A'B'C', with side lengths A'B' = (1/5)*AB, B'C' = (1/5)*BC, and A'C' = (1/5)*AC.
Now, let's calculate the perimeters of both triangles.
Perimeter of Triangle ABC = AB + BC + AC = 5 + 12 + 15
Perimeter of Triangle A'B'C' = A'B' + B'C' + A'C' = (1/5)*AB + (1/5)*BC + (1/5)*AC
To find the ratio of the perimeters, we can divide the perimeter of Triangle A'B'C' by the perimeter of Triangle ABC:
Ratio of the perimeters = (A'B' + B'C' + A'C') / (AB + BC + AC)
Substituting the values, we get:
Ratio of the perimeters = [(1/5)*AB + (1/5)*BC + (1/5)*AC] / (AB + BC + AC)
Simplifying this expression, we get:
Ratio of the perimeters = (1/5) * [(AB + BC + AC) / (AB + BC + AC)]
The numerator and denominator cancel out, leaving us with the answer:
Ratio of the perimeters = 1/5
Therefore, the ratio of the perimeters of the two triangles is 1/5, which means the perimeter of the new triangle (Triangle A'B'C') is 1/5 of the perimeter of the original triangle (Triangle ABC).
To determine the ratio of the perimeters of the original and new triangles, we first need to find the perimeters of both triangles.
1. Perimeter of the original triangle:
The original triangle has side lengths of 5, 12, and 15 inches. To find the perimeter, we add up these side lengths:
Perimeter = 5 + 12 + 15 = 32 inches
2. Perimeter of the new triangle:
The new triangle is formed by multiplying each side length of the original triangle by 1/5. Thus, the corresponding side lengths of the new triangle are (5/5), (12/5), and (15/5) inches. Now, we can calculate its perimeter:
New Perimeter = (5/5) + (12/5) + (15/5) = 32/5 = 6.4 inches
The ratio of the perimeters of the original and new triangles can be found by dividing the perimeters:
Ratio of perimeters = Perimeter of original triangle / Perimeter of new triangle
= 32 inches / 6.4 inches
= 5
Now let's consider the ratio of the corresponding sides of the original and new triangles:
1. Corresponding sides of the original triangle: 5, 12, and 15 inches
2. Corresponding sides of the new triangle: (5/5), (12/5), and (15/5) inches, which simplifies to 1, 2.4, and 3 inches respectively.
To find the ratio of the corresponding sides, we divide each side of the new triangle by the corresponding side of the original triangle:
Ratio of sides = (1/5) / 5, (2.4/5) / 12, (3/5) / 15
= 1/25, 12/24, 3/75
Simplifying the ratios, we get:
Ratio of sides = 1/25, 1/2, 1/25
The ratio of the perimeters (5) is equal to the ratio of the corresponding sides (1/25, 1/2, 1/25). In general, the ratio of the perimeters of similar figures is equal to the ratio of the corresponding sides.