The lengths of the sides of a triangle are 6 cm, 7 cm and 9 cm. In a similar triangle whose perimeter is 110 cm, what is the length of the longest side, in centimeters?
i know that the first triangle's perimeter is 22, but i'm not sure how this will help me.
I got the answer to my question. It was 45
To find the length of the longest side of a similar triangle whose perimeter is 110 cm, we can use the concept of similarity. Similar triangles have equal corresponding angles and proportional side lengths.
First, let's determine the ratio of the perimeters of the two triangles. The perimeter of the first triangle is 6 cm + 7 cm + 9 cm = 22 cm. The ratio of the perimeters is therefore given as:
Perimeter of the similar triangle / Perimeter of the first triangle = 110 cm / 22 cm
Simplifying this ratio, we get:
5
Since the perimeters are proportional, the corresponding side lengths of the similar triangle will also be proportional.
Next, we know that the longest side length in the first triangle is 9 cm. By using the proportionality ratio, we can find the length of the longest side in the similar triangle:
Longest side of the similar triangle / Longest side of the first triangle = Ratio of the perimeters
Substituting the values, we get:
Longest side of the similar triangle / 9 cm = 5
To find the length of the longest side in the similar triangle, we multiply both sides of the equation by 9 cm:
Longest side of the similar triangle = 5 * 9 cm
Simplifying, we find:
Longest side of the similar triangle = 45 cm
Therefore, the length of the longest side in the similar triangle is 45 cm.
To find the length of the longest side in the similar triangle, you need to find the scale factor between the perimeters of the two triangles.
Let's denote the scale factor as 'k' and the perimeter of the first triangle as 'P1' and the perimeter of the second triangle as 'P2'.
Since the perimeters of similar triangles are proportional, we can write:
P1 / P2 = k
Given that P1 (perimeter of the first triangle) is 22 cm and P2 (perimeter of the second triangle) is 110 cm, we can substitute these values into the equation:
22 / 110 = k
Simplifying, we get:
1 / 5 = k
Therefore, the scale factor 'k' is 1/5.
Now, let's find the length of the longest side in the second triangle. The longest side in the first triangle is 9 cm. Since the triangles are similar and the scale factor is 1/5, we can multiply the length of the longest side in the first triangle by the scale factor to find the length of the longest side in the second triangle:
Length of longest side in the second triangle = Length of longest side in the first triangle * Scale factor
Length of longest side in the second triangle = 9 cm * (1/5)
Simplifying, we get:
Length of longest side in the second triangle = 9/5 cm
Therefore, the length of the longest side in the similar triangle is 9/5 cm or 1.8 cm.