To find the coordinates of point $R$, where the line segment $QR$ is perpendicular to the line $1$, we can follow these steps:

Step 1: Identify the equation and gradient of line 1

Suppose the equation of line 1 is given in the slope-intercept form $y = mx + c$, where $m$ is the gradient (slope) of line 1, and $c$ is the y-intercept. If the equation of line 1 is not in slope-intercept form, we would first need to rearrange it into this form.

Step 2: Gradient of the line perpendicular to line 1

The gradient of a line perpendicular to line 1 is the negative reciprocal of the gradient of line 1. So, if the gradient of line 1 is $m$, then the gradient of line $QR$ (the perpendicular line) will be $-\frac{1}{m}$.

Step 3: Equation of the line through $Q$ and perpendicular to line 1

If the coordinates of point $Q$ are given as $(x_Q, y_Q)$, the equation of the line through point $Q$ with gradient $-\frac{1}{m}$ can be written as:

$y - y_Q = -\frac{1}{m}(x - x_Q)$

This is the point-slope form of the equation of line $QR$.

Step 4: Find the intersection of line 1 and line $QR$

The point $R$ lies on both line 1 and line $QR$, so we need to find the intersection of these two lines. To do this:

1. Substitute the equation of line $QR$ into the equation of line 1.
2. Solve for the coordinates of $R$.

Step 5: Solve for the coordinates of point $R$

Once we have the equation of line $QR$ and the equation of line 1, we can solve the system of equations to find the coordinates of point $R$.

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To summarize:

1. Determine the gradient of line 1.
2. Calculate the gradient of the perpendicular line $QR$.
3. Write the equation of line $QR$ passing through point $Q$.
4. Solve the system of equations to find the coordinates of $R$.

Example

Given:

• Line 1: $y = 2x + 3$ (where $m = 2$).
• Coordinates of $Q$: $(1, 5)$.

We can:

1. The gradient of line 1 is $m = 2$.
2. The gradient of line $QR$ is $-\frac{1}{2}$.
3. Write the equation of line $QR$ through $Q(1, 5)$:

$y - 5 = -\frac{1}{2}(x - 1)$

Simplifying this, we get:

$y - 5 = -\frac{1}{2}x + \frac{1}{2}$

$y = -\frac{1}{2}x + \frac{11}{2}$

4. To find the intersection with line 1, we set this equal to the equation of line 1:

$2x + 3 = -\frac{1}{2}x + \frac{11}{2}$

Now, solve for $x$ and $y$.

Let me know if you'd like a detailed breakdown of this example or if you have any other questions!