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NCERT solutions Mathematics Textbook for Class 10 chapter 7 Coordinate Geometry

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Chapter 7 - Coordinate Geometry

Pages 161 - 162

Find the distance between the following pair of point (2, 3), (4, 1)

 

Q 1.1 | Page 161 |

Find the distance between the following pairs of points: (−5, 7), (−1, 3)

Q 1.2 | Page 161 |

Find the distance between the following pair of point:(ab), (− a, − b)

Q 1.3 | Page 161 |

Find the distance between the points (0, 0) and (36, 15). Can you now find the distance between the two towns A and B discussed in Section 7.2.

Q 2 | Page 161 |

Check whether (5, - 2), (6, 4) and (7, - 2) are the vertices of an isosceles triangle.

Q 4 | Page 161 |

In a classroom, 4 friends are seated at the points A, B, C and D as shown in the following figure. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, “Don’t you think ABCD is a square?” Chameli disagrees.

Using distance formula, find which of them is correct.

Q 5 | Page 161 |

Name the type of quadrilateral formed, if any, by the given points, and give reasons for your answer: (- 1, - 2), (1, 0), (- 1, 2), (- 3, 0)

Q 6.1 | Page 161 |

Name the type of quadrilateral formed, if any, by the following point, and give reasons for your answer: (− 3, 5), (3, 1), (0, 3), (− 1, − 4)

Q 6.2 | Page 161 |

Name the type of quadrilateral formed, if any, by the following point, and give reasons for your answer (4, 5), (7, 6), (4, 3), (1, 2)

Q 6.3 | Page 161 |

Find the point on the x-axis which is equidistant from (2, - 5) and (- 2, 9).

 

Q 7 | Page 161 |

 Find the values of y for which the distance between the points P (2, - 3) and Q (10, y) is 10 units

 

Q 8 | Page 161 |

If Q (0, 1) is equidistant from P (5, − 3) and R (x, 6), find the values of x. Also find the distance QR and PR.

Q 9 | Page 162 |

Find a relation between x and y such that the point (xy) is equidistant from the point (3, 6) and (− 3, 4).

Q 10 | Page 162 |

Page 167

Find the coordinates of the point which divides the join of (–1, 7) and (4, –3) in the ratio 2 : 3.

Q 1 | Page 167 |

Find the coordinates of the points of trisection of the line segment joining (4, -1) and (-2, -3).

Q 2 | Page 167 |

To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1 m each. 100 flower pots have been placed at a distance of 1 m from each other along AD, as shown in the following figure. Niharika runs `1/4` th the distance AD on the 2nd line and posts a green flag. Preet runs `1/5`

th the distance AD on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?

Q 3 | Page 167 |

Find the ratio in which the line segment joining A (1, − 5) and B (− 4, 5) is divided by the x-axis. Also find the coordinates of the point of division.

Q 5 | Page 167 |

If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.

Q 6 | Page 167 |

Find the coordinates of a point A, where AB is the diameter of circle whose centre is (2, − 3) and B is (1, 4)

Q 7 | Page 167 |

If A and B are (− 2, − 2) and (2, − 4), respectively, find the coordinates of P such that `AP = 3/7 AB` and P lies on the line segment AB.

Q 8 | Page 167 |

Find the coordinates of the points which divide the line segment joining A (- 2, 2) and B (2, 8) into four equal parts.

Q 9 | Page 167 |

Find the area of a rhombus if its vertices are (3, 0), (4, 5), (− 1, 4) and (− 2, −1) taken in order. [Hint: Area of a rhombus = `1/2` (product of its diagonals)]

Q 10 | Page 167 |

Page 171

Determine the ratio in which the line 2x + y – 4 = 0 divides the line segment joining the points A(2, – 2) and B(3, 7).

Q 1 | Page 171 |

Find a relation between x and y if the points (x, y), (1, 2) and (7, 0) are collinear.

Q 2 | Page 171 |

 Find the centre of a circle passing through the points (6, − 6), (3, − 7) and (3, 3).

Q 3 | Page 171 |

The two opposite vertices of a square are (− 1, 2) and (3, 2). Find the coordinates of the other two vertices.

Q 4 | Page 171 |

The class X students of a secondary school in Krishinagar have been allotted a rectangular plot of land for their gardening activity. Saplings of Gulmohar are planted on the boundary at a distance of 1 m from each other. There is a triangular grassy lawn in the plot as shown in the following figure. The students are to sow seeds of flowering plants on the remaining area of the plot.

(i) Taking A as origin, find the coordinates of the vertices of the triangle.

(ii) What will be the coordinates of the vertices of Δ PQR if C is the origin?

Also calculate the areas of the triangles in these cases. What do you observe?

Q 5 | Page 171 |

The vertices of a ΔABC are A (4, 6), B (1, 5) and C (7, 2). A line is drawn to intersect sides AB and AC at D and E respectively, such that `(AD)/(AB) = (AE)/(AC) = 1/4`Calculate the area of the ΔADE and compare it with the area of ΔABC. (Recall Converse of basic proportionality theorem and Theorem 6.6 related to ratio of areas of two similar triangles)

Q 6 | Page 171 |

Let A (4, 2), B (6, 5) and C (1, 4) be the vertices of ΔABC.

(i) The median from A meets BC at D. Find the coordinates of point D.

(ii) Find the coordinates of the point P on AD such that AP: PD = 2:1

(iii) Find the coordinates of point Q and R on medians BE and CF respectively such that BQ: QE = 2:1 and CR: RF = 2:1.

(iv) What do you observe?

(v) If A(x1y1), B(x2y2), and C(x3y3) are the vertices of ΔABC, find the coordinates of the centroid of the triangle.

Q 7 | Page 171 |

ABCD is a rectangle formed by the points A(–1, –1), B(– 1, 4), C(5, 4) and D(5, – 1). P, Q, R and S are the mid-points of AB, BC, CD and DA respectively. Is the quadrilateral
PQRS a square? a rectangle? or a rhombus? Justify your answer.

Q 8 | Page 171 |

Extra questions

The coordinates of A, B, C are (6, 3), (–3, 5) and (4, – 2) respectively and P is any point (x, y). Show that the ratio of the areas of triangle PBC and ABC is



For what value of k are the points (k, 2 – 2k), (–k + 1, 2k) and (–4 – k, 6 – 2k) are collinear ?



Prove that the area of a triangle with vertices (t, t −2), (t + 2, t + 2) and (t + 3, t) is independent of t.



Prove that the points (2, – 2), (–3, 8) and (–1, 4) are collinear



For what value of x will the points (x, –1), (2, 1) and (4, 5) lie on a line ?



Prove that the points (a, b + c), (b, c + a) and (c, a + b) are collinear



The vertices of ∆ABC = are A (4, 6), B(1, 5) and C(7, 2). A line is drawn to intersect sides AB and AC at D and E respectively such that `\frac{AD}{AB}=\frac{AE}{AC}=\frac{1}{4}` .Calculate the area of ∆ADE and compare it with the area of ∆ABC



If A(4, –6), B(3, –2) and C(5, 2) are the vertices of ∆ABC, then verify the fact that a median of a triangle ABC divides it into two triangle of equal areas.



Find the lengths of the medians of a ∆ABC whose vertices are A(7, –3), B(5,3) and C(3,–1)



Find the area of the quadrilateral ABCD whose vertices are respectively A(1, 1), B(7, –3), C(12, 2) and D(7, 21).



If D, E and F are the mid-points of sides BC, CA and AB respectively of a ∆ABC, then using coordinate geometry prove that Area of ∆DEF = `\frac { 1 }{ 4 } "(Area of ∆ABC)"`



Find the area of the triangle whose vertices are: (8, 1), (k, -4), (2, -5)



Find the area of a triangle whose vertices are A(3, 2), B (11, 8) and C(8, 12).



Find the area of the triangle formed by joining the mid-point of the sides of the triangle whose vertices are (0, –1), (2, 1) and (0, 3). Find the ratio of area of the triangle formed to the area of the given triangle.



If the coordinates of two points A and B are (3, 4) and (5, – 2) respectively. Find the coordniates of any point P, if PA = PB and Area of ∆PAB = 10



Show that the points (a, a), (–a, –a) and (– √3 a, √3 a) are the vertices of an equilateral triangle. Also find its area.



If the point P(x, y) is equidistant from the points A(a + b, b – a) and B(a – b, a + b). Prove that bx = ay.



Find the value of x, if the distance between the points (x, – 1) and (3, 2) is 5.



Find the coordinates of the centre of the circle passing through the points (0, 0), (–2, 1) and (–3, 2). Also, find its radius.



If two vertices of an equilateral triangle be (0, 0), (3, √3 ), find the third vertex



Show that four points (0, – 1), (6, 7), (–2, 3) and (8, 3) are the vertices of a rectangle. Also, find its area



Prove that the points (–3, 0), (1, –3) and (4, 1) are the vertices of an isosceles right angled triangle. Find the area of this triangle



If P (2, – 1), Q(3, 4), R(–2, 3) and S(–3, –2) be four points in a plane, show that PQRS is a rhombus but not a square. Find the area of the rhombus



If the opposite vertices of a square are (1, – 1) and (3, 4), find the coordinates of the remaining angular points.



Find the coordinates of the circumcentre of the triangle whose vertices are (8, 6), (8, – 2) and (2, – 2). Also, find its circum radius



Show that the points (1, – 1), (5, 2) and (9, 5) are collinear.



Find the distance between two points

(i) P(–6, 7) and Q(–1, –5)

(ii) R(a + b, a – b) and S(a – b, –a – b)

(iii) `A(at_1^2,2at_1)" and " B(at_2^2,2at_2)`



If P and Q are two points whose coordinates are (at2 ,2at) and (a/t2 , 2a/t) respectively and S is the point (a, 0). Show that `\frac{1}{SP}+\frac{1}{SQ}` is independent of t.



If the coordinates of the mid points of the sides of a triangle are (1, 1), (2, – 3) and (3, 4) Find its centroid



If the points A (6, 1), B (8, 2), C(9, 4) and D(p, 3) are vertices of a parallelogram, taken in order, find the value of p



If the point C (–1, 2) divides internally the line segment joining A (2, 5) and B in ratio 3 : 4, find the coordinates of B



Find the coordinates of the centroid of a triangle whose vertices are (–1, 0), (5, –2) and (8, 2)



Two vertices of a triangle are (3, –5) and (–7, 4). If its centroid is (2, –1). Find the third vertex



Determine the ratio in which the line 3x + y – 9 = 0 divides the segment joining the points (1, 3) and (2, 7)



Prove that (4, – 1), (6, 0), (7, 2) and (5, 1) are the vertices of a rhombus. Is it a square?



Find the coordinates of points which trisect the line segment joining (1, –2) and (–3, 4)



The three vertices of a parallelogram taken in order are (–1, 0), (3, 1) and (2, 2) respectively. Find the coordinates of the fourth vertex.



Find the coordinates of the point which divides the line segment joining the points (6, 3) and (– 4, 5) in the ratio 3 : 2 internally.



If A(–2, –1), B(a, 0), C(4, b) and D(1, 2) are the vertices of a parallelogram, find the values of a and b



If A (5, –1), B(–3, –2) and C(–1, 8) are the vertices of triangle ABC, find the length of median through A and the coordinates of the centroid.



If the coordinates of the mid-points of the sides of a triangle are (1, 2) (0, –1) and (2, 1). Find the coordinates of its vertices.



Prove that the points (–2, –1), (1, 0), (4, 3) and (1, 2) are the vertices of a parallelogram. Is it a rectangle ?



In what ratio does the x-axis divide the line segment joining the points (2, –3) and (5, 6)? Also, find the coordinates of the point of intersection.



Prove that the diagonals of a rectangle bisect each other and are equal.



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