*Note: If the points are and , then the distance between them is equal to *

Question 1: Find the distance between the following pair of points:

(i) and

(ii) and

(iii) and

(iv) and

Answer:

(i) Distance between the given points and

(i) Distance between the given points and

(i) Distance between the given points and

(i) Distance between the given points and

Question 2: Find the value of when the distance between the points and is .

Answer:

Given:

Squaring both sides

Question 3: If the points and are equidistant from the point , show that .

Answer:

Given:

Question 4: Find the values of if the distances of the point from as well as from are .

Answer:

Given:

… … … … … i)

… … … … … ii)

From i) and ii) we get

Therefore from i),

Question 5: The length of a line segment is of units and the coordinates of one end-point are .If the abscissa of the other end is , find the ordinate of the other end.

Answer:

Given:

Question 6: Show that the points and are the vertices points of a rectangle.

Answer:

Let the points be and

Therefore sides of the figure

Diagonals of the figure

Hence, .Therefore the figure is a rectangle since the opposite sides are equal and the diagonals are also equal.

Question 7: Show that the points and are the vertices of a parallelogram.

Answer:

The points are and

Therefore sides of the figure

Diagonals of the figure

Hence, .Therefore the figure is a parallelogram since the opposite sides are equal but the diagonals are unequal.

Question 8: Prove that the points and are the vertices of a square.

Answer:

The points are and

Therefore sides of the figure

Diagonals of the figure

Hence, .Therefore the figure is a square since the opposite sides are equal and also the diagonals are equal.

Question 9: Prove that the points and are vertices of a right-angled isosceles triangle.

Answer:

Let the points be and

Therefore sides of the figure

We see that

Therefore is a right angled triangle at . Also therefore triangle is also an isosceles triangle.

Question 10: Prove that and are the vertices of a right angled triangle. Find the area of the triangle and the length of the hypotenuse.

Answer:

Let the points be and

Therefore sides of the figure

We see that

Therefore is a right angled triangle at .

Area sq. units

Hypotenuse

Question 11: Prove that the points and are the vertices of an equilateral triangle.

Answer:

Let the points be and

Therefore sides of the figure

We see that

Therefore is an equilateral triangle.

Question 12: Prove that the points and do not form a triangle.

Answer:

Let the points be and

Therefore sides of the figure

In a triangle, the sum of the length of to sides should be greater then the third side.

Because all three conditions are not TRUE, this is not a triangle.

Question 13: An equilateral triangle has two vertices at the points and , find the coordinates of the third vertex.

Answer:

Given vertices: and

Let the third vertices be

Also

Substituting

Therefore or

If

If

Therefore the coordinate of the third vertices would be

Question 14: Show that the quadrilateral whose vertices are and is a rhombus.

Answer:

Let the points be and

Therefore sides of the figure

Diagonals of the figure

Hence, . Therefore we see that the sides are equal but the diagonals are not equal. Hence is a Rhombus.

Question 15: Two vertices of an isosceles triangle are and . Find the third vertex if the length of the equal sides is .

Answer:

Given

Length of equal sides

Let the third vertices be

Therefore equal sides are and

… … … … … i)

… … … … … ii)

Subtracting i) from ii)

Therefore from i)

Therefore the coordinates are

Question 16: Which point on x-axis is equidistant from and ?

Answer:

Given and

Let be the equidistant point

Therefore

Therefore the point is

Question 17: Prove that the points and are collinear.

Answer:

Given

Therefore

Hence the points are collinear.

Question 18: The coordinates of the point are . Find the coordinates of the point which lies on the line joining and origin such that .

Answer:

Given . Let be

Now Slope of Slope of

Substituting

If

If

Since needs to be in the same line as , is

Question 19: Which point on y-axis is equidistant from and ?

Answer:

Given points

Let point be

Hence the point is

Question 20: The three vertices of a parallelogram are and . Find the fourth vertex.

Answer:

Given points

Let be

Since is a parallelogram

… … … … … i)

Also

… … … … … ii)

Slope of Slope of

Hence

Hence

Question 21: Find the circumcenter of the triangle whose vertices are .

Answer:

Given

Let be the circumcenter. Therefore

… … … … … i)

… … … … … ii)

Substituting i) in ii)

Hence

Therefore circumcenter is

Question 22: Find the angle subtended at the origin by the line segment whose end points are and .

Answer:

Given points and

Point is on y axis and point is on x axis.

Therefore angle subtended by and on origin is

Question 23: Find the center of the circle passing through and .

Answer:

Given

Let center be . Therefore

… … … … … i)

Substituting in i) we get

Hence the center of the circle is

Question 24: Find the value of , if the point is equidistant from and .

Answer:

Given

Since

Question 25: If two opposite vertices of a square arc and , find the coordinates of its remaining two vertices.

Answer:

Given .

Let be

Since is a square,

… … … … … i)

… … … … … ii)

Substituting i) into ii) we get

If then

If , then

Hence the other two corners of the square are and

Question 26: Show that the points and are the vertices of a rhombus. Find the area of this rhombus.

Answer:

Let the points be and

Therefore sides of the figure

Diagonals of the figure

Hence, . Therefore we see that the sides are equal but the diagonals are not equal. Hence is a Rhombus.

Area ( Product of lengths of diagonal)

sq. units

Question 27: Find the coordinates of the circumcenter of the triangle whose vertices are and . Also, find its circumradius.

Answer:

Given:

Let be the center

… … … … … i)

… … … … … ii)

Solving i) and ii) we get

Therefore

Hence the center is

Therefore Circumradius units

Question 28: Find a point on the x-axis which is equidistant from the points and .

Answer:

Given

Let the point be

Given

Therefore point is

Question 29: (i) show that the points and are the vertices of a square.

(ii) Prove that the points , and are the vertices of a square .

(iii) Name the type of formed by the points , and

Answer:

i) Let the points be and

Therefore sides of the figure

Diagonals of the figure

Hence, . Also diagonals are equal. Therefore the figure is a square.

ii) Let the points be and

Therefore sides of the figure

Diagonals of the figure

Hence, . Also diagonals are equal. Therefore the figure is a square.

iii) Let the points be

Therefore sides of the figure

Hence, . Therefore the figure is an equilateral triangle.

Question 30: Find the point on x-axis which is equidistant from the points and .

Answer:

Given

Let the equidistant point be

Therefore point is

Question 31: Find the value of such that where the coordinates of and are and respectively.

Answer:

Given:

Question 32: Prove, that the points and are the vertices of a right isosceles triangle.

Answer:

Let the points be

Therefore sides of the figure

Since is an Isosceles triangle.

Question 33: If the point is equidistant from the points and , prove that .

Answer:

Given:

Question 34: If is equidistant from and , find the values of . Also, find the distances and .

Answer:

Given:

When units

When units

Question 35: Find the values of for which the distance between the points and is units.

Answer:

Given:

Question 36: Find the center of the circle passing through and .

Answer:

Given:

Let center be

… … … … … i)

… … … … … ii)

Substituting in i)

Therefore center is

Question 37: Two opposite vertices of a square are and . Find the coordinates of other two vertices.

Answer:

Given

… … … … … i)

Also

From i),

Hence the coordinates are

Question 38: Name the quadrilateral formed, if any, by the following points, and give reasons for your answers:

(i)

(ii)

(iii)

Answer:

i) Let the points be and

Therefore sides of the figure

Diagonals of the figure

All four sides are equal and also the diagonals are equal. Hence this is a square.

ii) Let the points be and

Therefore sides of the figure

All four sides are unequal . Hence this is a quadrilateral.

iii) Let the points be and

Therefore sides of the figure

Diagonals of the figure

All four sides are equal and but the diagonals are unequal, this is a Rhombus.

Question 39: Find the equation of the perpendicular bisector of the line segment joining points and .

Answer:

Given:

Mid point of

Slope of

Therefore slope of perpendicular bisector

Therefore equation of perpendicular bisector is

Question 40: Prove that the points and , taken in order, form a rhombus. Also, find its area.

Answer:

Let the points be and

Therefore sides of the figure

Diagonals of the figure

All four sides are equal and but the diagonals are unequal, this is a Rhombus.

Area of the Rhombus (product of diagonals)

sq. units.

Question 41: In the seating arrangement of desks in a classroom three students Rohini, Sandhya and Bina are seated at and . Do you think they are seated in a line?

Answer:

Given:

Hence they are all in a straight line.

Question 42: Find a point on y-axis which is equidistant from the points and .

Answer:

Given:

Let the equidistant point be

Given

Therefore

Therefore the point is

Question 43: Find a relation between and such that the point is equidistant from the points and .

Answer:

Given

Let equidistant point

Given

Question 44: If a point is equidistant from the points and , then find the value of .

Answer:

Given:

Since

Question 45: Prove that the points and are the vertices of an isosceles right triangle.

Answer:

Given:

Therefore ( i.e. two sides of the triangle are equal)

Also

Therefore is a right angled isosceles triangle.

Question 46: If the point is equidistant from the points and , find the value of and find the distance .

Answer:

Give:

Since

Hence the point is

Question 47: It is equidistant from points and , find the value of and find the distance .

Answer:

Given:

Since

Therefore

Question 48: If and are the two vertices of an equilateral triangle, find the coordinates of its third vertex.

Answer:

Given:

Let third vertices be

Since

or … … … … … i)

Substituting in i)

Therefore could be or

Question 49: If the point is equidistant from the point s and , find . Also, find the length of .

Answer:

Given:

If

If

Question 50: If the point is equidistant from the points and , find . Also, find the length of .

Answer:

Given:

Therefore

Question 51: If the point is equidistant from the points and , find the values of .

Answer:

Given:

Question 52: If and are two vertices of an equilateral triangle, find the coordinates of the third vertex, given that the origin lies in the (i) interior, (ii) exterior of the triangle.

Answer:

Given:

Let third vertices be

Hence is i) and ii)

Question 53: Ayush starts walking from his house to office. Instead of going to the office directly, he goes to a bank first, from there to his daughter’s school and then reaches the office. What is the extra distance traveled by Ayush in reaching the office? (Assume that all distances covered are in straight hires). If the house is situated at , bank at school at and office at and coordinates are in kilometers.

Answer:

Given: House , Bank , School , Office

Distance from House to Office km

Distance of House to Bank km

Distance of Bank to School km

Distance from school to office km

Therefore total distance traveled km

Extra distance km

Question 54: The center of a circle is . Find the values of if the circle passes through the point and has diameter units.

Answer:

Given:

Radius

Therefore

Question 55: Find a point which is equidistant from the points and . How many such points are there?

Answer:

Given:

Let be equidistant

… … … … … i)

Therefore all points satisfying the equation i) will be equidistant from the two given points. Hence we have infinite such points.

Question 56: The points and are the vertices of a right angled at . Find the values of and hence the area of .

Answer:

Given:

When and hence

When and . Hence

Therefore Area sq units.