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: ln 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.

Link to the start of Exercise 19.4