For 2 similar triangles ABC and DEF, the scale factor of ABC to DEF is 2:3. If AB=2(BC) and DE=3(BC), what is EF?
I assume that you have listed the two triangles in matching vertices, that is, A <--> D , etc
let AB = 2x
then DE = 3x , must be in the ratio of 2:3
if AB = 2BC
2x = 2BC
BC = x
given: DE = 3BC
DE = 3x , (we already knew that)
then EF = 6x
Oh so we add 3X And the 3X to get 6X for EF ?
To find the length of EF, we can use the scale factor between the two triangles.
Given:
Scale factor of ABC to DEF = 2:3
AB = 2(BC)
DE = 3(BC)
Since the scale factor is 2:3, we can say that the ratio between AB and DE is also 2:3.
So, we have the ratio AB:DE = 2:3.
But we are given that AB = 2(BC) and DE = 3(BC).
Let's substitute these values into the ratio.
2(BC):3(BC) = 2:3
By canceling out the common factor of BC, we have:
2:3 = 2:3
Since the ratio is the same, it means that the sides are proportional.
Therefore, EF will also have the same proportion to BC.
Now, we know that BC is a common side for both triangles, so the ratio between the other sides of the triangles will also be 2:3.
Since BC is a common side, we can ignore it when determining the ratio of EF to DE.
Therefore, EF:DE = 2:3.
However, we are given that DE = 3(BC).
So, EF:3(BC) = 2:3.
To find EF, we can solve the equation by cross multiplying:
EF * 3(BC) = 2 * 3(BC)
3EF(BC) = 6(BC)
Dividing both sides by BC, we get:
3EF = 6
Dividing both sides by 3, we find:
EF = 2
Therefore, EF = 2.
To find the length of EF, we can use the scale factor between the triangles.
Given that the scale factor of ABC to DEF is 2:3, we know that the corresponding sides are in a ratio of 2:3.
Since AB = 2(BC), we can rewrite AB as 2x, where x represents the length of BC.
Similarly, DE = 3(BC) can be rewritten as DE = 3x.
Now, we can use the scale factor to find EF.
The scale factor of 2:3 tells us that AB is 2/3 of DE.
So, AB/DE = 2/3.
Substituting the lengths, we have 2x/3x = 2/3.
Now, we can solve for x by cross-multiplying.
2x(3) = 3x(2)
6x = 6x
x cancels out, so we are left with 6 = 6.
This means that x can be any value, as long as AB and DE are in the ratio of 2:3.
Since we are looking for the length of EF, we need to find DE first.
If DE = 3x, we substitute in any value of x.
Let's say we choose x = 1 for simplicity.
Then, DE = 3(1) = 3.
Now, we can find EF using the scale factor.
The scale factor tells us that AB is 2/3 of DE.
So, EF = (2/3)DE = (2/3)(3) = 2.
Therefore, EF is equal to 2.