R.S. Aggarwal solutions for Mathematics [English] Class 10 ICSE chapter 13 - Section and Mid-point Formulae [Latest edition]

Find the co-ordinates of point P which divides the line segment joining A(–2, –7) and B(6, 1) in the ratio 5 : 3.

The line segment joining the points \(\mathrm{A}(4, -3)\) and \(\mathrm{B}(4, 2)\) is divided by the point P such that \(\mathrm{AP}: \mathrm{AB}=2:5\). Find the co-ordinates of P.

\[[\textbf{Hint: } \frac{AP}{AB} = \frac{2}{5} \Rightarrow \frac{AP}{AP + PB} = \frac{2}{5} \Rightarrow 3AP = 2PB \Rightarrow \frac{AP}{PB} = \frac{2}{3} \Rightarrow AP : PB = 2 : 3\text{.]}\]

P(1, -2) is a point on the line segment A(3, -6) and B(x, y) such that AP : PB is equal to 2 : 3. Find the coordinates of B.

Find a point P on the line segment joining \(\mathrm{A}(14, -5)\) and \(\mathrm{B}(-4, 4)\), which is twice as far from A as from B. 

[Hint: \(AP = 2PB \Rightarrow AP : PB = 2 : 1\).]

Find the co-ordinates of the points of trisection of the line segment joining the points \(\mathrm{A}(5, -3)\) and \(\mathrm{B}(2, -9)\).

Find the co-ordinates of the mid-point of the line segment joining:

\(\mathrm{A}(5, 7)\) and \(\mathrm{B}(-3, -1)\)

Find the co-ordinates of the mid-point of the line segment joining:

\(\mathrm{P}(-5, -8)\) and \(\mathrm{Q}(3,4)\)

The line segment joining \(\mathrm{A}(-3, 1)\) and \(\mathrm{B}(7,-5)\) is a diameter of a circle whose centre is C. Find the co-ordinates of the point C.

\(\mathrm{A}(10, 5), \mathrm{B}(6, -3)\) and \(\mathrm{C}(2, 1)\) are the vertices of a \(\triangle \mathrm{ABC}\). \(\mathrm{L}\) is the mid-point of AB and M is the mid-point of AC. Write down the co-ordinates of L and M. Show that \(\mathrm{LM} = \frac{1}{2} \mathrm{BC}\).

The mid-point of the line segment joining \(\mathrm{A}(p, 5)\) and \(\mathrm{B}(3, q)\) is \(\mathrm{M}(-1, 4)\). Find the values of p and q.

The centre of a circle is C(–2, 3) and one end of a diameter PQ is P(2, –4). Find the co-ordinates of Q .

The point \(\mathrm{P}(-4, 1)\) divides the line segment joining the points \(\mathrm{A}(2, -2)\) and B in the ratio 3 : 5. Find the co-ordinates of point B.

In what ratio does the point P(2, –5) divide the join of A(–3, 5) and B(4, –9)?

In what ratio does the point \(\mathrm{P}(a, -1)\) divide the join of \(\mathrm{A}(1, -3)\) and \(\mathrm{B}(6, 2)\). Hence, find the value of a.

The line segment joining (\mathrm{A}(2, 3)) and (\mathrm{B}(6, -5)) is intercepted by the x-axis at the point k. Find the ratio in which k divides AB. Also, write the co-ordinates of the point k.

In what ratio is the segment joining the points \(\mathrm{A}(6,5)\) and \(\mathrm{B}(-3, 2)\) divided by the y-axis? Find the point at which the y-axis cuts AB.

1. Write down the co-ordinates of the point P that divides the line joining A(−4, 1) and B(17, 10) in the ratio 1 : 2.
2. Calculate the distance OP, where O is the origin.
3. In what ratio does the y-axis divide the line AB?

The line joining P(–4, 5) and Q(3, 2) intersects the y-axis at point R. PM and QN are perpendicular from P and Q on the x-axis Find:

1. the ratio PR : RQ
2. the coordinates of R.
3. the area of the quadrilateral PMNQ.

In the given figure, the line segment AB meets x-axis at A and y-axis at B. The point \(\mathrm{P}(-3, 1)\) on AB divides it in the ratio 2 : 3. Find the co-ordinates of A and B.
[Hint: (i) The co-ordinates of A and B are of the form A(x, 0), and B(0, y).]

Show that the line segment joining the points \(\mathrm{A}(-5, 8)\) and \(\mathrm{B}(10, -4)\) is trisected by the co-ordinate axes. Also, find the points of trisection of AB. 

[Hint: Let x-axis divide AB in the ratio k : 1 at the point P(x, 0). Let y-axis divide AB in the ratio p : 1 at the point Q(0, y).]

The mid-points of the sides BC, CA and AB of \(\Delta \mathrm{ABC}\) are \(\mathrm{D}(2, 1), \mathrm{E}(-1, -3)\) and \(\mathrm{F}(4, 5)\) respectively. Find the co-ordinates of \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\).

Prove that the points \(\mathrm{A}(-2, -1), \mathrm{B}(1, 0), \mathrm{C}(4, 3)\) and \(\mathrm{D}(1, 2)\) are the vertices of a parallelogram ABCD.

If the points A(–2, –1), B(1, 0), C(a, 3) and D(1, b) form a parallelogram, find the values of a and b.

[Hint: Mid-point of AC = mid-point of BD.]

The three vertices of a parallelogram ABCD, taken in order are \(\mathrm{A}(-1, 0), \mathrm{B}(3, 1)\) and \(\mathrm{C}(2, 2)\). Find the co-ordinates of the fourth vertex of the parallelogram.

Find the lengths of the medians of a \(\triangle \mathrm{ABC}\) whose vertices are \(\mathrm{A}(-1, 3), \mathrm{B}(1, -1)\) and \(\mathrm{C}(5, 1)\). Also, find the co-ordinates of the centroid of \(\Delta \mathrm{ABC}\).

Find the co-ordinates of the centroid of \(\triangle \mathrm{PQR}\) whose vertices are \(\mathrm{P}(6, 3), \mathrm{Q}(-2, 5)\) and \(\mathrm{R}(-1, 7)\).

Find the co-ordinates of the point of intersection of the medians of the triangle whose vertices are A(–7, 5), B(–1, –3) and C(5, 7).

[Hint. Find the centroid of \(\triangle A B C\).]

If \(\mathrm{G}(-2, 1)\) is the centroid of \(\triangle \mathrm{ABC}\), two of whose vertices are \(\mathrm{A}(1, -6)\) and \(\mathrm{B}(-5, 2)\), find the third vertex of the triangle.

A(6, y), B(–4, 4) and C(x, –1) are the vertices of \(\triangle \mathrm{ABC}\) whose centroid is the origin. Calculate the values of x and y.

ABC is a triangle and \(\mathrm{G}(4, 3)\) is its centroid. If \(\mathrm{A}(1, 3), \mathrm{B}(4, b)\) and \(\mathrm{C}(a, 1)\) be the vertices, find the values of a and b and hence find the length of side BC.

Calculate the ratio in which the line segment joining \(\mathrm{A}(-4, 2)\) and \(\mathrm{B}(3, 6)\) is divided by the point \(\mathrm{P}(x, 3)\). Also, find (i) x (ii) length of AP.

In what ratio is the line joining \(\mathrm{P}(5, 3)\) and \(\mathrm{Q}(-5, 3)\) divided by the x-axis? Also find the coordinates of the point of intersection.

Find a point P which divides internally the line segment joining the points A(–3, 9) and B(1, –3) in the ratio 1 : 3.

The coordinates of the point P which divides the join of A(5, –2) and B(9, 6) in the ratio 3 : 1 are ______.

The coordinates of the point on x-axis which divides the line segment joining the points (2, 3) and (5, – 6) in the ratio 1 : 2 are ______.

The point which divides the line segment joining the points A(3, –2) and B(6, 7) internally in the ratio 3 : 2 lies in which of the following quadrants?

If the point \(\mathrm{R}(k, 4)\) divides the line segment joining the points \(\mathrm{P}(2,6)\) and \(\mathrm{Q}(5, 1)\) in the ratio 2 : 3, then the value of (k) is ______.

Points \(\mathrm{A}(x, y), \mathrm{B}(3, -2)\) and \(\mathrm{C}(4, -5)\) are collinear. The value of y in terms of x is ______.

If the point \(\mathrm{P}(6, 2)\) divides the line segment joining \(\mathrm{A}(6, 5)\) and \(\mathrm{B}(4, y)\) in the ratio 3 : 1, then the value of y is ______.

The ratio in which the point \(\mathrm{P}(1, 2)\) divides the join of the points \(\mathrm{A}(-2, 1)\) and \(\mathrm{B}(7, 4)\) is ______.

The line segment joining \(\mathrm{A}(-7, 2)\) and \(\mathrm{B}(3, -8)\) is divided by the x-axis in the ratio: 

In what ratio is the line segment joining the points \(\mathrm{P}(-4, 2)\) and \(\mathrm{Q}(8, 3)\) divided by y-axis?

Point P divides the line segment joining R(-1, 3) and S(9,8) in ratio k:1. If P lies on the line x - y + 2 = 0, then value of k is ______.

In the adjoining figure, \(\mathrm{P}(5, -3)\) and \(\mathrm{Q}(3, y)\) are the points of trisection of the line segment joining \(\mathrm{A}(7, -2)\) and \(\mathrm{B}(1, -5)\). Then, y equals:  

If the point \(\mathrm{P}(6, -3)\) lies on the line segment joining points \(\mathrm{A}(4, 2)\) and \(\mathrm{B}(8, 4)\) then:

The mid-point of the line segment joining the points (–3, 2) and (7, 6) is ______.

If A(4, 2), B(6, 5) and C(1, 4) be the vertices of \(\triangle \mathrm{ABC}\) and AD is a median, then the coordinates of D are ______.

If (3, – 6) is the mid-point of the line segment joining (0, 0) and (x,  y), then the point (x, y) is ______.

If the line segment joining the points P and Q(3, –4) is bisected at the origin, then the coordinate of P are ______.

\(\mathrm{A}(-3, b)\) and \(\mathrm{B}(1, b + 4)\) are two points. If the coordinates of the mid-point of AB are \((-1, 1)\), then the value of b is ______.

If the point \(\mathrm{R}(5,7)\) is the mid-point of the line segment joining the points \(\mathrm{P}(3, y)\) and \(\mathrm{Q}(x, 9)\), then \((x + y)\) equals:

The mid-point of the line segment joining (4p, 5) and (2, 3q) is (5, 5p – 1). The values of p and q are respectively are ______.

A line intersects the y-axis and x-axis at the points P and Q respectively. If (2, –5) is the mid-point of PQ, then the coordinates of P and Q are respectively:

If a point \(\mathrm{R}\left(\frac{23}{5}, \frac{33}{5}\right)\) divides the line segment PQ joining the points \(\mathrm{P}(3, 5)\) and \(\mathrm{Q}(x, y)\) in the ratio 2 : 3 internally, then the values of x and y respectively are ______.

The coordinates of the vertices of ΔABC are respectively (–4, –2), (6, 2), and (4, 6). The centroid G of ΔABC is ______.

Two vertices of a \(\triangle \mathrm{ABC}\) are \(\mathrm{A}(-1,4)\) and \(\mathrm{B}(5, 2)\) and its centroid is (0, –3). The coordinates of C are ______.

In the adjoining diagram, G is the centroid of ΔABC. A(3, –3), В(2, –6), C(x, y) and G(5, –5). The coordinates of point D are:

The points \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) divide the line segment joining the points \(\mathrm{P}(-3, 8)\) and \(\mathrm{Q}(9, -4)\) into four equal parts. If A is nearest to P, then the coordinates of A are ______.

The line 2x + y – 4 = 0 divides the line segment joining \(\mathrm{A}(2, -2)\) and \(\mathrm{B}(3, 7)\) in the ratio

The centre of the circle having end points of its one diameter as (–4, 2) and (4, –3), is ______.

A circle has its centre at (4, 4). If one end of a diameter is (4, 0), then the coordinates of the other end are ______.

The vertices of a parallelogram in order are A(1, 2), B(4, y), C(x, 6) and D(3, 5). Then (x, y) is ______.

The fourth vertex D of a parallelogram ABCD, whose three vertices are A(–2, 3), B(6, 7) and C(8, 3), is ______.

\(\mathrm{A}(1,4), \mathrm{B}(4,1)\) and \(\mathrm{C}(x, 4)\) are the vertices of \(\triangle \mathrm{ABC}\). If the centroid of the triangle is \(\mathrm{G}(4, 3)\), then x is equal to ______.

Assertion (A): The coordinates of a point which divides a line segment joining the points (–3, 4) and (7, –6) in the ratio 1 : 2 internally are \(\left(\frac{1}{3}, \frac{2}{3}\right)\).

Reason (R): The coordinates of the point which divides the line segment. joining the points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) internally in the ratio \(m : n\) are given by \(\left(\frac{mx_{2} + nx_{1}}{m + n}, \frac{my_{2} + ny_{1}}{m + n}\right)\)

Assertion (A): The coordinates of one end of a diameter of a circle are (1, 4). If its centre is at (2, –3), then the coordinates of the other end of the diameter are (–3, –10).

Reason (R): Centre of a circle is equidistant from each end of a diameter.

Assertion (A): If the points \(\mathrm{A}(x, 1), \mathrm{B}(8, 2), \mathrm{C}(9, 4)\) and \(\mathrm{D}(7, 3)\) are the vertices of a parallelogram, taken in order, then the value of x is 6.

Reason (R): Adjacent angles of a parallelogram are supplementary.

Assertion (A): The coordinates of the centroid of a triangle whose vertices are (–1, –4), (4, 3) and (6 – 2) are (3, –1).

Reason (R): If the vertices of a triangle are \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right)\), then the coordinates of its centroid are \(\left(\frac{x_{1} + x_{2} + x_{3}}{2}, \frac{x_{1} + x_{2} + x_{3}}{2}\right)\)

Determine the ratio in which the line y = 2 + 3x divides the line segment AB joining the points A(–3, 9) and B(4, 2).

In the given figure, if the line segment AB is intercepted by the y-axis and x-axis at C and D, respectively, such that AC : AD = 1 : 4 and D is the midpoint of CB. Find the coordinates of D, C and В.