CLASS X : CHAPTER - 7 COORDINATE GEOMETRY

CLASS X : CHAPTER - 7 COORDINATE GEOMETRY

IMPORTANT FORMULAS & CONCEPTS

Points to remember
The distance of a point from the y-axis is called its x-coordinate, or abscissa.
The distance of a point from the x-axis is called its y-coordinate, or ordinate.
The coordinates of a point on the x-axis are of the form (x, 0).
The coordinates of a point on the y-axis are of the form (0, y).

Distance Formula
The distance between any two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:
\( AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
or \( AB = \sqrt{(\text{difference of abscissae})^2 + (\text{difference of ordinates})^2} \)
Distance of a point from origin: The distance of a point \( P(x, y) \) from origin O is given by \( OP = \sqrt{x^2 + y^2} \).

Problems based on geometrical figures
To show that a given figure is a:

• Parallelogram: prove that the opposite sides are equal.
• Rectangle: prove that the opposite sides are equal and the diagonals are equal.
• Parallelogram but not rectangle: prove that the opposite sides are equal and the diagonals are not equal.
• Rhombus: prove that the four sides are equal.
• Square: prove that the four sides are equal and the diagonals are equal.
• Rhombus but not square: prove that the four sides are equal and the diagonals are not equal.
• Isosceles triangle: prove any two sides are equal.
• Equilateral triangle: prove that all three sides are equal.
• Right triangle: prove that sides of triangle satisfy Pythagoras theorem.

Section Formula
The coordinates of the point \( P(x, y) \) which divides the line segment joining the points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), internally, in the ratio \( m_1 : m_2 \) are:
\( \left( \frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \frac{m_1y_2 + m_2y_1}{m_1 + m_2} \right) \)

Mid-point Formula
The coordinates of the point \( P(x, y) \) which is the midpoint of the line segment joining the points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), are:
\( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)

Area of a Triangle
If \( A(x_1, y_1) \), \( B(x_2, y_2) \) and \( C(x_3, y_3) \) are the vertices of a \( \Delta ABC \), then the area of \( \Delta ABC \) is given by:
\( \text{Area of } \Delta ABC = \frac{1}{2} [x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)] \)

MCQ WORKSHEET-I

1. The points A(0, –2), B(3, 1), C(0, 4) and D(–3, 1) are the vertices of a:
   1. parallelogram
   2. rectangle
   3. square
   4. rhombus
2. If A(3, 8), B(4, –2) and C(5, –1) are the vertices of \( \Delta ABC \). Then, its area is:
   1. \( 28 \frac{1}{2} \) sq. units
   2. \( 37 \frac{1}{2} \) sq. units
   3. 57 sq. units
   4. 75 sq. units
3. Two vertices of \( \Delta ABC \) are A(–1, 4) and B(5, 2) and its centroid is G(0, –3). The coordinate of C is:
   1. (4, 3)
   2. (4, 15)
   3. (–4, –15)
   4. (–15, –4)
4. If the points A(2, 3), B(5, k) and C(6, 7) are collinear, then the value of k is:
   1. 4
   2. 6
   3. \( -\frac{3}{2} \)
   4. \( \frac{11}{4} \)
5. x–axis divides the join of A(2, –3) and B(5, 6) in the ratio:
   1. 3 : 5
   2. 2 : 3
   3. 2 : 1
   4. 1 : 2
6. What point on x – axis is equidistant from the points A(7, 6) and B(–3, 4)?
   1. (0, 4)
   2. (–4, 0)
   3. (3, 0)
   4. (0, 3)
7. The distance of the point P(4, –3) from the origin is:
   1. 1 unit
   2. 7 units
   3. 5 units
   4. 3 units

MCQ WORKSHEET-II

1. Find the area of the triangle whose vertices are A(10, –6), B(2, 5) and C(–1, 3):
   1. 12.5 sq. units
   2. 24.5 sq. units
   3. 7 sq. units
   4. 6.5 sq. units
2. For what value of x are the points A(–3, 12), B(7, 6) and C(x, 9) collinear?
   1. 1
   2. –1
   3. 2
   4. –2
3. What is the midpoint of a line with endpoints (–3, 4) and (10, –5)?
   1. (–13, –9)
   2. (–6.5, –4.5)
   3. (3.5, –0.5)
   4. none of these
4. The fourth vertex of the rectangle whose three vertices taken in order are (4,1), (7, 4), (13, –2) is:
   1. (10, –5)
   2. (10, 5)
   3. (8, 3)
   4. (8, –3)
5. If four vertices of a parallelogram taken in order are (–3, –1), (a, b), (3, 3) and (4, 3). Then a : b =
   1. 1 : 4
   2. 4 : 1
   3. 1 : 2
   4. 2 : 1
6. If the origin is the mid-point of the line segment joined by the points (2,3) and (x,y), then the value of (x,y) is:
   1. (2, –3)
   2. (2, 3)
   3. (–2, 3)
   4. (–2, –3)

MCQ WORKSHEET-III

1. The distance of the point P(2, 3) from the x-axis is:
   1. 2
   2. 3
   3. 1
   4. 5
2. The distance of the point P(-6, 8) from the origin is:
   1. 8
   2. 27
   3. 10
   4. 6
3. AOBC is a rectangle whose three vertices are A(0, 3), O(0, 0) and B(5, 0). The length of its diagonal is:
   1. 5
   2. 3
   3. \( \sqrt{34} \)
   4. 4
4. The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is:
   1. 5
   2. 12
   3. 11
   4. 7 + \sqrt{5}
5. Line formed by joining (- 1,1) and (5, 7) is divided by a line x + y = 4 in the ratio of:
   1. 1 : 4
   2. 1 : 3
   3. 1 : 2
   4. 3 : 4

MCQ WORKSHEET-IV

1. If the points (1, x), (5, 2) and (9, 5) are collinear then the value of x is:
   1. \( \frac{5}{2} \)
   2. \( -\frac{5}{2} \)
   3. –1
   4. 1
2. The ratio in which x – axis divides the line segment joining the points (5, 4) and (2, –3) is:
   1. 5 : 2
   2. 3 : 4
   3. 2 : 5
   4. 4 : 3
3. The point which lies on the perpendicular bisector of the line segment joining the points A(–2, –5) and B(2, 5) is:
   1. (0, 0)
   2. (0, 2)
   3. (2, 0)
   4. (–2, 0)
4. Three vertices of a parallelogram taken in order are (- 1, - 6), (2, - 5) and (7, 2). The fourth vertex is:
   1. (1, 4)
   2. (1, 1)
   3. (4, 4)
   4. (4, 1)
5. The area of the triangle with vertices at the points (a, b + c), (b, c + a) and (c, a + b) is:
   1. a + b + c
   2. a + b – c
   3. a – b + c
   4. 0

PRACTICE QUESTIONS: DISTANCE FORMULA

1. Find the distance between the following points:
   (i) A(9, 3) and (15, 11)
   (ii) A(7, – 4) and b(–5, 1).
   (vi) \( P(a \sin\alpha, a \cos\alpha) \) and \( Q(a \cos\alpha, -a \sin\alpha) \)
2. If A(6, –1), B(1, 3) and C(k, 8) are three points such that AB= BC, find the value of k.
3. Find all the possible values of a for which the distance between the points A(a, –1) and B(5, 3) is 5 units.
4. Determine if the points (1, 5), (2, 3) and (– 2, – 11) are collinear.
5. Find the point on x-axis which is equidistant from (–2, 5) and (2, –3).
6. Find a point on the y-axis which is equidistant from the points A(6, 5) and B(– 4, 3).
7. Prove that the points A(a, a), B(–a, –a) and \( C(-\sqrt{3}a, \sqrt{3}a) \) are the vertices of an equilateral triangle. Calculate the area of this triangle.
8. Show that the points A(1, 2), B(5, 4), C(3, 8) and D(–1, 6) are vertices of a square.
9. Show that the points A(1, 0), B(5, 3), C(2, 7) and D(–2, 4) are vertices of a rhombus.
10. Find the coordinates of the circumcentre of a triangle whose vertices are A(4, 6), B(0, 4) and C(6, 2). Also, find its circumradius.
11. Find the coordinates of the centre of a circle passing through the points A(2, 1), B(5, –8) and C(2, –9). Also find the radius of this circle.
12. If the points A(6, 1), B(8, 2), C(9, 4) and D(p, 3) are the vertices of a parallelogram, taken in order, find the value of p.
13. Find a relation between x and y such that the point (x , y) is equidistant from the points (7, 1) and (3, 5).
14. If the point P(x, y) is equidistant from the points A(5, 1) and B(–1, 5), prove that x = y.

PRACTICE QUESTIONS: SECTION FORMULA

1. Find the coordinates of the point which divides the line segment joining the points A(4, –3) and B(9, 7) in the ratio 3 : 2.
2. Find the coordinates of the midpoint of the line segment joining the points A(–5, 4) and B(7, –8).
3. Find the coordinates of the points which divide the line segment joining the points (–2, 2) and (2, 8) in four equal parts.
4. In what ratio does the points P(2,–5) divide the line segment joining A(–3, 5) and B(4, –9).
5. Find the coordinates of the points of trisection of the line segment joining the points (4, –1) and (–2, –3).
6. The line segment joining the points (3, –4) and (1, 2) is trisected at the points P(p, –2) and \( Q(\frac{5}{3}, q) \). Find the values of p and q.
7. Find the ratio in which the y-axis divides the line segment joining the points (5, – 6) and (–1, – 4). Also find the point of intersection.
8. Find the ratio in which the line segment joining the points (– 3, 10) and (6, – 8) is divided by (– 1, 6).
9. Find the ratio in which the line 2x + y – 4 = 0 divides the line segment joining the points A(2, – 2) and B(3, 7).
10. If the point C(–1, 2) divides the line segment AB in the ratio 3 : 4, where the coordinates of A are (2, 5), find the coordinates of B.
11. Find the centroid of \( \Delta ABC \) whose vertices are A(–3, 0), B(5, –2) and C(–8, 5).
12. If the points (10, 5), (8, 4) and (6, 6) are the midpoints of the sides of a triangle, find its vertices.

PRACTICE QUESTIONS: AREA OF TRIANGLE

1. Find the area of a triangle formed by the points A(5, 2), B(4, 7) and C(7, – 4).
2. Show that the points \( (-\frac{3}{2}, 3) \), (6, –2), (–3, 4) are collinear by using area of triangle.
3. By using area of triangle show that the points (a, b + c), (b, c + a) and (c, a + b) are collinear.
4. Find the value of k if the points A(8, 1), B(k, –4) and C(2, –5) are collinear.
5. If A(–5, 7), B(– 4, –5), C(–1, –6) and D(4, 5) are the vertices of a quadrilateral, find the area of the quadrilateral ABCD.
6. Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, –1), (2, 1) and (0, 3). Find the ratio of this area to the area of the given triangle.
7. If the vertices of a triangle are (1, k), (4, –3) and (–9, 7) and its area is 15 sq. units, find the value(s) of k.
8. Find the value of k for which the points with coordinates (2, 5), (4, 6) and \( (k, \frac{11}{2}) \) are collinear.
9. Prove that the area of triangle whose vertices are (t, t – 2), (t + 2, t + 2) and (t + 3, t) is independent of t.
10. If the coordinates of two points A and B are (3, 4) and (5, –2) respectively. Find the coordinates of any point P, if PA = PB and area of \( \Delta PAB \) = 10 sq. units.
11. If three points (a, b), (c, d) and (e, f) are collinear, prove that \( d(e - a) + f(a - c) + b(c - e) = 0 \).