Answer:
If two triangles are mathematically similar, their corresponding sides are proportional, and their areas are proportional to the square of the corresponding sides.
Let's say that the corresponding sides of triangles ABC and DEF are:
AB:DE = BC:EF = AC:DF = k
Then the ratio of their areas is:
(area of ABC)/(area of DEF) = (ABBC)/2 / (DEEF)/2 = (AB/DE)*(BC/EF) = k^2
We know that the area of triangle ABC is 34 cm², so:
(area of ABC)/(area of DEF) = 34/(area of DEF) = k^2
We need to find the length of triangle DEF, which is the same as finding the value of k. To do that, we need to find a pair of corresponding sides. Let's say that the length of AB is 4 cm, and the length of BC is 17 cm.
Then, using the ratio of corresponding sides, we can find the lengths of DE and EF:
DE = (AB/DE)DE = k4
EF = (BC/EF)EF = k17
Now we can find the area of triangle DEF:
(area of DEF) = (area of ABC) / (k^2) = 34 / (k^2)
(area of DEF) = (DEEF)/2 = (k4)(k17)/2 = 34k²
Now we can equate the two expressions for the area of triangle DEF:
34/(k^2) = 34k²
Dividing both sides by 34 gives:
1/(k^2) = k²
Taking the square root of both sides gives:
k = 1
Therefore, the corresponding sides of triangles ABC and DEF are equal in length, and the area of triangle DEF is also 34 cm²