Answer:
PQ is 10.
QR is 5.
Hence, PQ=2QR
Step-by-step explanation:
We have the three points P(-1, 3); Q(7, -3); and R(4, 1).
And we want to show that PQ=2QR.
In other words, we want to show that PQ/QR=2.
So, let's find PQ and QR. We will need to use the distance formula:
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2[/tex]
To find PQ:
P is (-1, 3) and Q is (7, -3).
So, we will let P(-1, 3) be (x₁, y₁) and Q(7, -3) be (x₂, y₂).
Substitute the values into the distance formula. This yields:
[tex]d=\sqrt{(7-(-1))^2+(-3-3)^2[/tex]
Evaluate:
[tex]d=\sqrt{(8)^2+(-6)^2[/tex]
Evaluate:
[tex]d=\sqrt{64+36}=\sqrt{100}=10[/tex]
So, the distance of PQ is 10.
And to find QR:
Q is (7, -3) and R is (4, 1).
Again, we will let Q(7, -3) be (x₁, y₁) and R(4, 1) be (x₂, y₂).
Substitute appropriately. So:
[tex]d=\sqrt{(4-7)^2+(1-(-3))^2[/tex]
Evaluate:
[tex]d=\sqrt{(-3)^2+(4)^2[/tex]
Evaluate:
[tex]d=\sqrt{9+16}=\sqrt{25}=5[/tex]
So, the distance of QR is 5.
Therefore, it follows that:
[tex]\displaystyle PQ=2QR\Rightarrow \frac{PQ}{QR}=2\Rightarrow\frac{10}{5}\stackrel{\checmark}{=}2[/tex]
And we have shown that PQ=2QR.
