Question

P(-,3) , Q(7,-3) and R (4,1) are three points. show that PQ=2QR using distance formula. (please show the steps too :)​

Answer 1

PQ is 10.

QR is 5.

Hence, PQ=2QR

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:

`d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2`

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:

`d=\sqrt{(7-(-1))^2+(-3-3)^2`

Evaluate:

`d=\sqrt{(8)^2+(-6)^2`

Evaluate:

`d=√(64+36)=√(100)=10`

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:

`d=\sqrt{(4-7)^2+(1-(-3))^2`

Evaluate:

`d=\sqrt{(-3)^2+(4)^2`

Evaluate:

`d=√(9+16)=√(25)=5`

So, the distance of QR is 5.

Therefore, it follows that:

`\displaystyle PQ=2QR\Rightarrow (PQ)/(QR)=2\Rightarrow(10)/(5)\stackrel{\checmark}{=}2`

And we have shown that PQ=2QR.