- Identify the coordinates of points \(P(5,7)\), \(Q(2,3)\), and the general form of \(R(x_R, 1)\).
- Calculate the slope of the segment \(PQ\) as \(\frac{4}{3}\) and determine that the slope of the perpendicular segment \(QR\) must be \(-\frac{3}{4}\).
- Set up an algebraic equation for the slope of \(QR\) using the coordinates of \(Q\) and \(R\) and solve for \(x_R\).
- Conclude that the x-coordinate of point \(R\) is \(\boxed{\frac{14}{3}}\).
Explanation
- Analyze the geometry and given conditions.
The problem asks to find the x-coordinate of point \(R\), given that triangle \(PQR\) has a right angle at vertex \(Q\), with \(Q = (2, 3)\), \(P = (5, 7)\), and point \(R\) is located on the line \(y = 1\).
Since \(R\) lies on the line \(y = 1\), its coordinates can be written as \((x_R, 1)\).
The condition that the angle \(\angle PQR\) is a right angle implies that the line segment \(PQ\) is perpendicular to the line segment \(QR\). This means the product of their slopes must be \(-1\), provided the segments are not vertical or horizontal.
- Calculate the slopes of the segments.
First, we find the slope of the segment \(PQ\), denoted by \(m_{PQ}\). Using the coordinates of \(P(5, 7)\) and \(Q(2, 3)\):
\[m_{PQ} = \frac{y_P - y_Q}{x_P - x_Q} = \frac{7 - 3}{5 - 2} = \frac{4}{3}\]
Since \(PQ \perp QR\), the slope of the segment \(QR\), denoted by \(m_{QR}\), is the negative reciprocal of \(m_{PQ}\):
\[m_{QR} = -\frac{1}{m_{PQ}} = -\frac{1}{4/3} = -\frac{3}{4}\]
- Solve for the x-coordinate of R.
Now, we express the slope of segment \(QR\) using the coordinates of \(Q(2, 3)\) and \(R(x_R, 1)\):
\[m_{QR} = \frac{y_R - y_Q}{x_R - x_Q} = \frac{1 - 3}{x_R - 2} = \frac{-2}{x_R - 2}\]
We equate this expression to the slope we found in the previous step:
\[\frac{-2}{x_R - 2} = -\frac{3}{4}\]
To solve for \(x_R\), we cross-multiply:
\[(-2)(4) = (-3)(x_R - 2)\]
\[-8 = -3x_R + 6\]
Adding \(3x_R\) to both sides and adding 8 to both sides:
\[3x_R = 6 + 8\]
\[3x_R = 14\]
\[x_R = \frac{14}{3}\]
Examples
In navigation or surveying, understanding the relationship between perpendicular paths is essential. If a surveyor maps out a path \(PQ\) to a central node \(Q\), and needs to set a second node \(R\) such that the path \(QR\) is perpendicular to \(PQ\) to form a right-angled boundary for a plot of land, this geometric approach allows them to calculate the exact location of node \(R\) on a specific line. By knowing the coordinates of two points, they can ensure the layout is perfectly square and accurate.