Verify shape type from coordinates

6 \includegraphics[max width=\textwidth, alt={}, center]{a10ad459-6f86-4845-ba28-e4e394bf3d1e-3_602_570_260_790} In the diagram, triangle \(A B C\) is right-angled and \(D\) is the mid-point of \(B C\). Angle \(D A C = 30 ^ { \circ }\) and angle \(B A D = x ^ { \circ }\). Denoting the length of \(A D\) by \(l\),

1. express each of \(A C\) and \(B C\) exactly in terms of \(l\), and show that \(A B = \frac { 1 } { 2 } l \sqrt { } 7\),
2. show that \(x = \tan ^ { - 1 } \left( \frac { 2 } { \sqrt { } 3 } \right) - 30\).

3 \includegraphics[max width=\textwidth, alt={}, center]{933cdfe1-27bb-450d-8b9a-b494916242cb-2_737_693_1484_726} In the diagram, \(A B E D\) is a trapezium with right angles at \(E\) and \(D\), and \(C E D\) is a straight line. The lengths of \(A B\) and \(B C\) are \(2 d\) and \(( 2 \sqrt { 3 } ) d\) respectively, and angles \(B A D\) and \(C B E\) are \(30 ^ { \circ }\) and \(60 ^ { \circ }\) respectively.

1. Find the length of \(C D\) in terms of \(d\).
2. Show that angle \(C A D = \tan ^ { - 1 } \left( \frac { 2 } { \sqrt { 3 } } \right)\).

4 The coordinates of the points \(\mathrm { A } , \mathrm { B }\) and C are ( \(- 2,2\) ), ( 1,3 ) and ( \(3 , - 3\) ) respectively.

1. Find the gradients of the lines AB and BC .
2. Show that the triangle ABC is a right-angled triangle.
3. Find the area of the triangle ABC .

5 The vertices of a triangle have coordinates ( 1,5 ), ( \(- 3,7\) ) and ( \(- 2 , - 1\) ).
Show that the triangle is right-angled.

12 ABCD is a parallelogram. The coordinates of \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D are (-2, 3), (2, 4), (8, -3) and ( \(4 , - 4\) ) respectively. \includegraphics[max width=\textwidth, alt={}, center]{3b6291ef-bef9-49de-a20f-591e621bed65-4_592_725_387_492}

1. Prove that AB and BD are perpendicular.
2. Find the lengths of AB and BD and hence find the area of the parallelogram ABCD
3. Find the equation of the line CD and show that it meets the \(y\)-axis at \(\mathrm { X } ( 0 , - 5 )\).
4. Show that the lines BX and AD bisect each other.
5. Explain why the area of the parallelogram ABCD is equal to the area of the triangle BXC.
   Find the length of BX and hence calculate exactly the perpendicular distance of C from BX .

14

1. In Fig. C1.3, angle CBD \(= \theta\). Show that angle CDA is also \(\theta\), as given in line 23 .
2. Prove that \(h = \sqrt { a b }\), as given in line 24 .

2 The triangle \(A B C\) has vertices \(A ( 1,3 ) , B ( 3,7 )\) and \(C ( - 1,9 )\).

1. 1. Find the gradient of \(A B\).
   2. Hence show that angle \(A B C\) is a right angle.
2. 1. Find the coordinates of \(M\), the mid-point of \(A C\).
   2. Show that the lengths of \(A B\) and \(B C\) are equal.
   3. Hence find an equation of the line of symmetry of the triangle \(A B C\).

\includegraphics{figure_10} Fig. 10 is a sketch of quadrilateral ABCD with vertices A \((1, 5)\), B \((-1, 1)\), C \((3, -1)\) and D \((11, 5)\).

1. Show that \(AB = BC\). [3]
2. Show that the diagonals AC and BD are perpendicular. [3]
3. Find the midpoint of AC. Show that BD bisects AC but AC does not bisect BD. [5]

\includegraphics{figure_8} Fig. 10 is a sketch of quadrilateral ABCD with vertices A \((1, 5)\), B \((-1, 1)\), C \((3, -1)\) and D \((11, 5)\).

1. Show that AB = BC. [3]
2. Show that the diagonals AC and BD are perpendicular. [3]
3. Find the midpoint of AC. Show that BD bisects AC but AC does not bisect BD. [5]

Points \(A(-7, -7)\), \(B(8, -1)\), \(C(4, 9)\) and \(D(-11, 3)\) are the vertices of a quadrilateral \(ABCD\).

1. Prove that \(ABCD\) is a rectangle. [4 marks]
2. Find the area of \(ABCD\). [2 marks]