A triangle whose sides are 5,12, and 13 inches is similar to a traingle whose longest side is 39 inches. What is the perimeter of the larger triangle?
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To determine the perimeter of the larger triangle, we need to find the lengths of its other two sides.
Since the smaller triangle is similar to the larger triangle, their corresponding sides are proportional.
Let's determine the scale factor by comparing the lengths of the longest sides:
Scale factor = (Length of longest side of larger triangle ÷ Length of longest side of smaller triangle)
Scale factor = (39 inches ÷ 13 inches) = 3
Now, using the scale factor, we can find the lengths of the other two sides of the larger triangle:
Length of one side of larger triangle = (Length of corresponding side of smaller triangle) × (Scale factor)
Length of one side of larger triangle = 5 inches × 3 = 15 inches
Length of the second side of the larger triangle is not given, but we know it is proportional to the second side of the smaller triangle. Using the scale factor:
Length of the second side of larger triangle = (Length of second side of smaller triangle) × (Scale factor)
Length of the second side of larger triangle = 12 inches × 3 = 36 inches
Now we can find the perimeter of the larger triangle by summing the lengths of all three sides:
Perimeter of the larger triangle = Longest side + Side 1 + Side 2
Perimeter of the larger triangle = 39 inches + 15 inches + 36 inches = 90 inches
Therefore, the perimeter of the larger triangle is 90 inches.
To solve this problem, we need to understand the concept of similarity. Two triangles are similar when their corresponding angles are equal, and their corresponding sides are proportional. In this case, the two triangles are similar because they share equal angles.
Given that the sides of the smaller triangle are 5, 12, and 13 inches, we can deduce that it is a right-angled triangle with sides in the ratio 5:12:13. This is actually a Pythagorean triple, where the hypotenuse is equal to twice the smallest side. Hence, the smallest side is 5 inches, the second side is 12 inches, and the longest side (the hypotenuse) is 13 inches.
Now, we can determine the proportion between the sides of the two triangles. The longest side of the smaller triangle is 13 inches, while the longest side of the larger triangle is 39 inches. Therefore, we can set up the proportion:
13/39 = x/13
Simplifying, we get:
1/3 = x/13
To find the value of `x`, we can cross-multiply:
3x = 13
x = 13/3
Therefore, the smallest side of the larger triangle is 13/3 inches.
To find the perimeter of the larger triangle, we need to add up the lengths of all three sides. The sides of the larger triangle are in proportion to those of the smaller triangle, so we can simply multiply each side by the same ratio `x`:
Smallest side: (13/3) inches
Second side: 12 * (13/3) inches
Longest side: 39 inches
Perimeter = (13/3) + 12 * (13/3) + 39
Now, we can calculate the perimeter of the larger triangle by simplifying the expression.
the perimeter of two similar triangles is proportional to the ratio of their sides
ratio of sides = 13:39 = 1:3
the perimeter of the smaller is 30
so the perimeter of the larger is 90.