Name the type of triangle formed by the points \(A(-5,6),\ B(-4,-2),\ C(7,5)\).
Scalene.
Find the point(s) on the x–axis at a distance \(2\sqrt5\) from \((7,-4)\). How many such points are there?
(5, 0) and (9, 0) — two points.
What type of quadrilateral is formed by the points \(A(2,-2),\ B(7,3),\ C(11,-1),\ D(6,-6)\) in order?
Rectangle.
Find \(a\) if the distance between \(A(-3,-14)\) and \(B(a,-5)\) is \(9\).
\(a=-3\)
Find a point equidistant from \(A(-5,4)\) and \(B(-1,6)\). How many such points are there?
One example is (-3, 5). There are infinitely many such points (on the perpendicular bisector).
Find the coordinates of the point \(Q\) on the x–axis which lies on the perpendicular bisector of the segment joining \(A(-5,-2)\) and \(B(4,-2)\). Also, name the type of triangle formed by \(Q, A, B\).
\(Q = \left(-\dfrac{1}{2}, 0\right)\)
Triangle \(QAB\) is isosceles (\(QA=QB\)).
Find \(m\) if the points \((5,1),\ (-2,-3),\ (8,2m)\) are collinear.
\(m=\dfrac{19}{14}\)
If \(A(2,-4)\) is equidistant from \(P(3,8)\) and \(Q(-10,y)\), find \(y\). Also find \(PQ\).
\(y=-3\) or \(y=-5\)
\(PQ=\sqrt{290}\) when \(y=-3\); and \(PQ=13\sqrt{2}\) when \(y=-5\).
Find the area of the triangle with vertices \((-8,4),\ (-6,6),\ (-3,9)\).
0
In what ratio does the x–axis divide the segment joining \((-4,-6)\) and \((-1,7)\)? Find the coordinates of the point of division.
Ratio \(6:7\) (internally), point \(\big(-\dfrac{34}{13},\,0\big)\).
Find the ratio in which the point \(P\big(\dfrac{3}{4},\dfrac{5}{12}\big)\) divides the segment joining \(A\big(\dfrac{1}{2},\dfrac{3}{2}\big)\) and \(B(2,-5)\).
\(1:5\) (\(AP:PB\)).
If \(P(9a-2,-b)\) divides the segment joining \(A(3a+1,-3)\) and \(B(8a,5)\) in the ratio \(3:1\), find \(a\) and \(b\).
\(a=1,\ b=-3\)
If \((a,b)\) is the midpoint of the segment joining \(A(10,-6)\) and \(B(k,4)\) and \(a-2b=18\), find \(k\) and \(|AB|\).
\(k=22\), \(|AB|=2\sqrt{61}\)
The centre of a circle is \((2a,\,a-7)\). If it passes through \((11,-9)\) and has diameter \(10\sqrt2\), find \(a\).
\(a=3\) or \(a=5\)
The line segment joining the points \(A(3,2)\) and \(B(5,1)\) is divided at the point \(P\) in the ratio \(1:2\), and \(P\) lies on the line \(3x - 18y + k = 0\). Find the value of \(k\).
\(k=19\)
If \(D\big(-\dfrac{1}{2},\dfrac{5}{2}\big),\ E(7,3),\ F\big(\dfrac{7}{2},\dfrac{7}{2}\big)\) are the midpoints of the sides of \(\triangle ABC\), find \(\text{area}(\triangle ABC)\).
\(11\) square units.
The points \(A(2,9),\ B(a,5),\ C(5,5)\) are vertices of a triangle right–angled at \(B\). Find \(a\) and the area of \(\triangle ABC\).
\(a=2\), area \(=6\).
Find the coordinates of the point \(R\) on the segment joining \(P(-1,3)\) and \(Q(2,5)\) such that \(PR=\dfrac{3}{5}PQ\).
\(R\big(\dfrac{4}{5},\dfrac{21}{5}\big)\)
Find \(k\) if the points \(A(k+1,2k),\ B(3k,2k+3),\ C(5k-1,5k)\) are collinear.
\(k=2\)
Find the ratio in which the line \(2x+3y-5=0\) divides the segment joining \(B(8,-9)\) and \(C(2,1)\). Also find the coordinates of the point of division.
Ratio \(8:1\) (internally, from \(B:C\)). Point \(\big(\dfrac{8}{3},\,-\dfrac{1}{9}\big)\).