Perpendicular from point to line

Find the equation of a perpendicular from a point to a line, or find where this perpendicular meets the line.

9 questions · Moderate -0.2

1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships
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CAIE P1 2008 June Q11
9 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{d71002bb-b6f0-42a3-89fb-f2769d5c3779-4_563_965_813_591} In the diagram, the points \(A\) and \(C\) lie on the \(x\) - and \(y\)-axes respectively and the equation of \(A C\) is \(2 y + x = 16\). The point \(B\) has coordinates ( 2,2 ). The perpendicular from \(B\) to \(A C\) meets \(A C\) at the point \(X\).
  1. Find the coordinates of \(X\). The point \(D\) is such that the quadrilateral \(A B C D\) has \(A C\) as a line of symmetry.
  2. Find the coordinates of \(D\).
  3. Find, correct to 1 decimal place, the perimeter of \(A B C D\).
CAIE P1 2013 June Q7
8 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-3_465_554_255_794} The diagram shows three points \(A ( 2,14 ) , B ( 14,6 )\) and \(C ( 7,2 )\). The point \(X\) lies on \(A B\), and \(C X\) is perpendicular to \(A B\). Find, by calculation,
  1. the coordinates of \(X\),
  2. the ratio \(A X : X B\).
CAIE P1 2016 June Q11
12 marks Standard +0.3
11 Triangle \(A B C\) has vertices at \(A ( - 2 , - 1 ) , B ( 4,6 )\) and \(C ( 6 , - 3 )\).
  1. Show that triangle \(A B C\) is isosceles and find the exact area of this triangle.
  2. The point \(D\) is the point on \(A B\) such that \(C D\) is perpendicular to \(A B\). Calculate the \(x\)-coordinate of \(D\).
CAIE P1 2006 November Q5
7 marks Moderate -0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-3_684_771_260_685} The three points \(A ( 1,3 ) , B ( 13,11 )\) and \(C ( 6,15 )\) are shown in the diagram. The perpendicular from \(C\) to \(A B\) meets \(A B\) at the point \(D\). Find
  1. the equation of \(C D\),
  2. the coordinates of \(D\).
OCR MEI C1 Q5
4 marks Moderate -0.8
5 A line has gradient - 4 and passes through the point (2,6). Find the coordinates of its points of intersection with the axes. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{01569a16-66ba-422e-a74d-6f9430dd245b-3_433_835_353_715} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} Fig. 11 shows the line joining the points \(\mathrm { A } ( 0,3 )\) and \(\mathrm { B } ( 6,1 )\).
  1. Find the equation of the line perpendicular to AB that passes through the origin, O .
  2. Find the coordinates of the point where this perpendicular meets AB .
  3. Show that the perpendicular distance of AB from the origin is \(\frac { 9 \sqrt { 10 } } { 10 }\).
  4. Find the length of AB , expressing your answer in the form \(a \sqrt { 10 }\).
  5. Find the area of triangle OAB .
AQA Paper 1 2021 June Q5
6 marks Moderate -0.3
5
  1. Find the equation of the line perpendicular to \(L\) which passes through \(P\). 5 The line \(L\) has equation 5
  2. Hence, find the shortest distance from \(P\) to \(L\). \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-05_2488_1716_219_153}
OCR MEI C1 2009 June Q11
12 marks Moderate -0.3
\includegraphics{figure_11} Fig. 11 shows the line joining the points A \((0, 3)\) and B \((6, 1)\).
  1. Find the equation of the line perpendicular to AB that passes through the origin, O. [2]
  2. Find the coordinates of the point where this perpendicular meets AB. [4]
  3. Show that the perpendicular distance of AB from the origin is \(\frac{9\sqrt{10}}{10}\). [2]
  4. Find the length of AB, expressing your answer in the form \(a\sqrt{10}\). [2]
  5. Find the area of triangle OAB. [2]
Edexcel C1 Q10
13 marks Moderate -0.3
The straight line \(l_1\) has equation \(2x + y - 14 = 0\) and crosses the \(x\)-axis at the point \(A\).
  1. Find the coordinates of \(A\). [2]
The straight line \(l_2\) is parallel to \(l_1\) and passes through the point \(B(-6, 6)\).
  1. Find an equation for \(l_2\) in the form \(y = mx + c\). [3]
The line \(l_2\) crosses the \(x\)-axis at the point \(C\).
  1. Find the coordinates of \(C\). [1]
The point \(D\) lies on \(l_1\) and is such that \(CD\) is perpendicular to \(l_1\).
  1. Show that \(D\) has coordinates \((5, 4)\). [5]
  2. Find the area of triangle \(ACD\). [2]
OCR C1 Q10
13 marks Moderate -0.3
The straight line \(l_1\) has equation \(2x + y - 14 = 0\) and crosses the \(x\)-axis at the point \(A\).
  1. Find the coordinates of \(A\). [2]
The straight line \(l_2\) is parallel to \(l_1\) and passes through the point \(B(-6, 6)\).
  1. Find an equation for \(l_2\) in the form \(y = mx + c\). [3]
The line \(l_2\) crosses the \(x\)-axis at the point \(C\).
  1. Find the coordinates of \(C\). [1]
The point \(D\) lies on \(l_1\) and is such that \(CD\) is perpendicular to \(l_1\).
  1. Show that \(D\) has coordinates \((5, 4)\). [5]
  2. Find the area of triangle \(ACD\). [2]