NCERT Class 10 Chapter 4 Coordinate Geometry Official CBSE Board Sample Problems Short Answer

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Short Answer

Question

If the point P(x, y) is equidistant from the points and Prove that .

Solution

Equidistant

Equidistant

Equidistant

Given,

Applying distance formula

Hence proved

Question

Prove that the points (2, -2), (-2, 1) and (5, 2) are the vertices of a right angled triangle. Also find the area of this triangle.

Solution

Right angled triangle.

Right Angled Triangle.

Right angled triangle.

Let and be the given points. So,

Using Distance formula

, so is a right angled triangle in which BC is hypotenuse.

sq. Units

Question

Find the ratio in which the point (-3, k) divides the line-segment joining the points (-5, -4) and (-2, 3). Also find the value of k.

Solution

The line-segment

The Line-Segment

The line-segment

Let P divides AB in k: 1.

Then (Using section formula, ]

Hence the required ratio is 2: 1

Question

Prove that the points (30), (6, 4) and (-1, 3) are the vertices of a right angled isosceles triangle.

Solution

Isosceles triangle.

Isosceles Triangle.

Isosceles triangle.

Let the triangle be ABC as shown in figure. Distances are:

Using distance formula.

Here, is isosceles triangle

Consider,

and,

Here,

is a right angled triangle.

: In right & using Pythagoras theorem

Where H = hypotenuse. B = base, P = perpendiculars

Question

Let P and Q be the points of trisection of the line segment joining the points and such that P is nearer to A. Find the coordinates of P and Q.

Solution

The line segment

The Line Segment

The line segment

Let be given points. Let are point of trisection.

P divides AB in the ratio l :2

Coordinates of P are

Q is midpoint of PB. So using midpoint formula coordinates of are or

Question

If the point divides internally the line-segment joining the points and in the ratio 3: 4, find the value of .

Solution

The line segment

The Line Segment

The line segment

Using section formula,

Similarly, 2

Ilcncc,

Question

In figure ABC is a triangle coordinates of whose vertex A are D and E respectively are the mid-points of the sides AB and AC and their coordinates are (1, 0) and (0, 1) respectively. If F is the mid-point of BC, find the areas of MBC and DEF.

Solution

Triangle coordinates

Triangle Coordinates

Triangle coordinates

Let coordinates of B are . Then using midpoint formula we

Coordinates of B are (2, l) 2

Let coordinates of C are (p, q)

Similarly coordinates of C we have

Coordinates of C are (0, 3)

Area of

sq. units

Coordinates of F are i.e. (1, 2)( Using mid-point formula we

Area of =

sq. units (Area cannot he negative)

Question

Find the coordinates of the points which divide the line segment joining and into four equal parts.

Solution

Let points P, Q and R divide AB in to 4 equal parts.

Question

In what ratio does the y-axis divide the line segment joining the points and?

Also, find the coordinates of the point of intersection.

Solution

----------------------

Hence y axis divides PQ in the ratio 4: 3

Coordinates of point Pare

Question

Prove that the points (a, 0), (0, b) and (1, 1) are collinear if,

Solution

If the points are collinear, area of triangle form by these three points is equal to zero.

Area of the triangle

Divide both side by ab

Hence

Question

If the points and form a parallelogram, find the values of x and y.

Solution

Let the vertices of a parallelogram taken in order be and. Since diagonals of a parallelogram bisect each other,

Midpoint of AC= Midpoint of BD

and

Question

Find a relation between x and y such that the point is equidistant from the points and

Solution

Let be equidistant from the points and

Given,

By distance formula,

Question

Layout of a park

Layout of a Park

Layout of a park

The figure shows layout of a park. In the region TQRS roses are planted which needs to be fenced. Find the length of wire required to fence the rose bed?

Solution

Similarly,

Question

In what ratio does the line divides the join of and ?

Solution

Let divide the join of P and Q in the ratio k: 1

Question

Prove that the points and are collinear.

Solution

Area =

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