triangle abc is similar to def, the lengths of the sides of triangle abc is 5,8,11. what is the length of the shortest side of triangle def if its perimeter is 60?
If similar, then their sides are in the same ratio, so ..
let the sides of triangle DEF be
5x , 8x, and 11x
5x + 8x + 11x = 60
24x = 60
x = 60/24 = 2.5
so the shortest side is 5x = 5(2.5) = 12.5
YEWALASET
Ah, geometry! Keeping shapes similar but changing their sizes, just like my ever-changing jokes! Now, if triangle ABC, with sides 5, 8, and 11, is similar to triangle DEF, and DEF has a perimeter of 60, we can solve this. Since the sides are directly proportional, we can find the ratio of the perimeters:
Perimeter ratio = (DEF perimeter) / (ABC perimeter)
Perimeter ratio = 60 / (5 + 8 + 11)
Now, if we multiply this ratio by each side length of triangle ABC, we can find the corresponding side lengths of triangle DEF. So, let's calculate:
Shortest side of DEF = (Shortest side of ABC) * (Perimeter ratio)
Shortest side of DEF = 5 * (60 / 24)
Now, let me calculate that for you:
Shortest side of DEF = 12.5
Therefore, the length of the shortest side of triangle DEF, when its perimeter is 60, would be 12.5.
To find the length of the shortest side of triangle DEF, we need to use the fact that triangle ABC is similar to triangle DEF.
The ratio of the lengths of corresponding sides of similar triangles is equal.
Given that the lengths of the sides of triangle ABC are 5, 8, and 11, we can determine the corresponding sides of triangle DEF using the proportion:
AB / DE = BC / EF = AC / DF
Let's calculate the proportion using the given information:
AB / DE = 5 / x
BC / EF = 8 / y
AC / DF = 11 / z
Since the perimeter of triangle DEF is given as 60, we can set up the equation:
DE + EF + DF = 60
Substituting the proportions into the equation, we have:
5 / x + 8 / y + 11 / z = 60
We don't have enough information to solve for the individual side lengths, but we can determine the length of the shortest side by finding the corresponding ratio:
AB / DE = 5 / x
Since AB is the shortest side of triangle ABC with a length of 5, we can solve for x by setting up the proportion:
5 / x = AB / DE
Substituting the given side lengths:
5 / x = 5 / DE
Cross-multiplying:
5 * DE = 5 * x
Dividing both sides by 5:
DE = x
Therefore, the length of the shortest side of triangle DEF is x, which is equal to the length of side AB of triangle ABC, which is 5 units.
To solve this problem, we need to use the property of similar triangles. Since triangle ABC is similar to triangle DEF, the corresponding sides of both triangles are proportional.
We know the lengths of the sides of triangle ABC: AB = 5, BC = 8, and CA = 11. We need to find the length of the shortest side of triangle DEF.
To find the ratio of the corresponding sides between the two triangles, we can use the formula:
Ratio = Length of corresponding side in DEF / Length of corresponding side in ABC
Let's assume the length of the shortest side in DEF is x. Therefore, the ratio of the shortest side in DEF to the shortest side in ABC is x / 5.
We also know that the perimeter of DEF is 60. The perimeter of a triangle is the sum of all its sides. So, we can form an equation using the side lengths of DEF:
x + (length of second side in DEF) + (length of third side in DEF) = 60
Since we are looking for the length of the shortest side in DEF, we need to find the length of the other two sides. To do this, we can use the ratio of the side lengths of ABC to DEF.
Let's set up the equation using the ratio:
x / 5 + (8/5) * (length of second side in ABC) + (11/5) * (length of third side in ABC) = 60
Now, substitute the known values of the side lengths of ABC into the equation:
x / 5 + (8/5) * 8 + (11/5) * 11 = 60
Simplify the equation:
x / 5 + (64/5) + (121/5) = 60
Combine the fractions:
x / 5 + (185/5) = 60
Now, bring the fractions on one side and the whole numbers on the other side:
x / 5 = 60 - (185/5)
Simplify the right side of the equation:
x / 5 = 300/5 - 185/5
x / 5 = 115/5
Now, multiply both sides of the equation by 5 to isolate x:
x = (115/5) * 5
x = 115
Therefore, the length of the shortest side of triangle DEF is 115 units.