*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

Link to the rest of Exercise 19.4