a line segment has end point P(3,6) and Q(12,18) and is dilated so that's its new endpoints are Q' (8,12). What is the scale factor? If the lenght of PQ is 15 what is the lenght of P' Q'?
Can you plez help worth 10 points
the scale Q'/Q = 2/3
2/3 * 15 = 10, the new length
25
P'Q' has length = 10, so scale factor is 10/15 = 2/3
Sure, I'd be happy to help you with this question! To find the scale factor, we can compare the coordinates of the original endpoints (P and Q) to the coordinates of the new endpoint (Q').
Let's start by finding the distance between P and Q:
Distance PQ = √((x₂ - x₁)² + (y₂ - y₁)²)
P(x₁, y₁) = (3, 6)
Q(x₂, y₂) = (12, 18)
Distance PQ = √((12 - 3)² + (18 - 6)²)
= √(9² + 12²)
= √(81 + 144)
= √225
= 15
So, the length of PQ is 15 units.
Now, let's find the distance between P' and Q':
P'(x₁', y₁') = (3', 6')
Q'(x₂', y₂') = (8, 12)
We can assume that the scale factor is the same for both x and y coordinates. Let's call it "k".
x₁' = x₁ * k
y₁' = y₁ * k
3' = 3 * k (equation 1)
6' = 6 * k (equation 2)
Similarly, for Q':
x₂' = x₂ * k
y₂' = y₂ * k
8 = 12 * k (equation 3)
12 = 18 * k (equation 4)
Now, let's solve equations 3 and 4 to find the value of k:
From equation 3, we have:
k = 8/12
k = 2/3
Now, let's use this value of k in equations 1 and 2 to find the coordinates of P':
3' = 3 * (2/3)
= 2
6' = 6 * (2/3)
= 4
So, the coordinates of P' are (2, 4).
Now, we can find the length of P'Q' using the same formula as before:
Distance P'Q' = √((x₂' - x₁')² + (y₂' - y₁')²)
= √((8 - 2)² + (12 - 4)²)
= √(6² + 8²)
= √(36 + 64)
= √100
= 10
Therefore, the length of P'Q' is 10 units.
To summarize, the scale factor is 2/3 and the length of P'Q' is 10 units.