We know that dilations preserve angle measures, which means that the corresponding angles of the triangles are congruent. Let's review our diagram, paying special attention to the angles of the triangles.
Since we know that ∠ N ≅ ∠ N', we can use the two measures find the value of y.
y=2y-60 ⇒ y=60
Finding the Scale Factor
To find the value of x, we first must determine the scale factor of the dilation image. Since we know that the point L' is on PL, we can use the measures of PL and PL' to find the scale factor n. Let's look at our figure again.
We can see that the length of PL is 2 and the length of PL' is 4. We can use this information to calculate the scale factor. Remember that the image length is always in the numerator of the scale factor.
n=PL'/PL = 4/2 = 2
Therefore, the scale factor of the given dilation is n=2.
Finding the Value of x
The scale factor is the ratio of any length of the image to the corresponding length in the preimage. Let's review our diagram one more time.
We can use the scale factor n=2 to find the value of x. We see that L'M'=x+3 and LM=x. Let's substitute these values into our scale factor formula and solve for x.
