The smaller of two similar triangles has a perimeter of 20cm (#a+b+c=#20cm). The lengths of the longest sides of both triangles are in proportion 2:5. What is the perimeter of the larger triangle? Please explain.

Question

Violet Aldrich · Accepted Answer

#color(white)(xx)50# #color(white)(xx)a+b+c=20# Let sides of larger triangle are #a'#, #b'#, and #c'#. If similarity proportion is #2/5#, then, #color(white)(xx)a'=5/2a#, #color(white)(xx)b'=5/2b#, and#color(white)(x)c'=5/2c# #=>a'+b'+c'=5/2(a+b+c)# #=>a'+b'+c'=5/2color(red)(*20)# #color(white)(xxxxxxxxxxx)=50#

Ella Armstrong · Answer

undefined To find the perimeter of the larger triangle, we need to determine the lengths of its sides. Since the triangles are similar, their corresponding sides are in proportion.

Let the lengths of the longest sides of the smaller and larger triangles be 2x and 5x, respectively. Since the perimeter of the smaller triangle is 20 cm, the sum of its sides' lengths (a + b + c) is 20 cm.

Given that the longest side of the smaller triangle is 2x, we can write its perimeter as 2x + (other two sides).

Since the triangles are similar, the ratio of corresponding sides is the same. Therefore, if the longest side of the smaller triangle is 2x, the longest side of the larger triangle is 5x.

Now, the perimeter of the smaller triangle can be written as:
2x + (other two sides) = 20 cm

Since the longest side of the larger triangle is 5x, its perimeter can be written as:
5x + (other two sides of the larger triangle)

We know that the sum of the perimeters of both triangles is the same. Therefore, we can set up the equation:
2x + (other two sides of the smaller triangle) = 5x + (other two sides of the larger triangle)

Since the sum of the sides of each triangle is its perimeter, we can equate the perimeter of the smaller triangle to 20 cm:
2x + (other two sides of the smaller triangle) = 20

Now, we can solve for the "other two sides" of the smaller triangle:
other two sides of the smaller triangle = 20 - 2x

Substituting this expression into the equation for the perimeter of the larger triangle, we get:
5x + (20 - 2x) = Perimeter of the larger triangle

Solving this equation will give us the perimeter of the larger triangle.