Coordinate Geometry Exercise 7.4

NCERT solutions for Maths Coordinate Geometry

NCERT Solutions for Class 10 Maths Exercise 7.4

NCERT Solutions for Class 10 Maths Coordinate Geometry

1. Determine t]“1he ratio in which the line divides the line segment joining the points A and B

Ans. Let the line divides the line segment joining A and B (3, 7) in the ratio at point C. Then, the coordinates of C are

But C lies on , therefore

Hence, the required ratio if 2: 9 internally.

NCERT Solutions for Class 10 Maths Exercise 7.4

2. Find a relation between and if the points and are collinear.

Ans. The points A B (1, 2) and C (7, 0) will be collinear if

Area of triangle = 0

NCERT Solutions for Class 10 Maths Exercise 7.4

3. Find the centre of a circle passing through the points and

Ans. Let P be the centre of the circle passing through the points A B and C (3, 3). Then AP = BP = CP.

Taking AP = BP

……….(i)

Again, taking BP = CP

Putting the value of in eq. (i),

Hence, the centre of the circle is

NCERT Solutions for Class 10 Maths Exercise 7.4

4. The two opposite vertices of a square are and Find the coordinates of the other two vertices.

Ans. Let ABCD be a square and B be the unknown vertex.

AB = BC

……….(i)

In ABC,

……….(ii)

Putting the value of in eq. (ii),

= 0 or 4

Hence, the required vertices of the square are (1, 0) and (1, 4).

NCERT Solutions for Class 10 Maths Exercise 7.4

5. 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 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 PQR if C is the origin? Also calculate the area of the triangle in these cases. What do you observe?

Ans. (i) Taking A as the origin, AD and AB as the coordinate axes. Clearly, the points P, Q and

R are (4, 6), (3, 2) and (6, 5) respectively.

(ii)Taking C as the origin, CB and CD as the coordinate axes. Clearly, the points P, Q and R are given by (12, 2), (13, 6) and (10, 3) respectively.

We know that the area of the triangle =

Area of PQR (First case) =

= = sq. units

And Area of PQR (Second case) =

= = sq. units

Hence, the areas are same in both the cases.

NCERT Solutions for Class 10 Maths Exercise 7.4

6. The vertices of a 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 Calculate the area of the ADE and compare it with the area of ABC.

Ans. Since,

DE BC[By Thales theorem]

ADE ABC

……….(i)

Now, Area (ABC) =

= sq. units……….(ii)

From eq. (i) and (ii),

Area (ADE) = Area (ABC) = sq. units

Area (ADE): Area (ABC) = 1: 16

NCERT Solutions for Class 10 Maths Exercise 7.4

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

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

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

(iii) Find the coordinates of points 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?

(Note: The point which is common to all the three medians is called centroid and this point divides each median in the ratio 2: 1)

(v) If A B and C are the vertices of ABC, find the coordinates of the centroid of the triangle.

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

(i) Since AD is the median of ABC.

D is the mid-point of BC.

Its coordinates are =

(ii) Since P divides AD in the ratio 2: 1

Its coordinates are =

(iii) Since BE is the median of ABC.

E is the mid-point of AD.

Its coordinates are =

Since Q divides BE in the ratio 2: 1.

Its coordinates are =

Since CF is the median of ABC.

F is the mid-point of AB.

Its coordinates are =

Since R divides CF in the ratio 2: 1.

Its coordinates are =

(iv) We observe that the points P, Q and R coincide, i.e., the medians AD, BE and CF are concurrent at the point . This point is known as the centroid of the triangle.