Question 1: If the line segment joining the points and subtends an angle at the origin , prove that: .

Answer:

Consider the figure given.

Using cosine formula in we get

Question 2: The vertices of a and and . Find

Answer:

Given

We know where

Question 3: Four points ) and are given in such a way that , find .

Answer:

Given ) and

We know that the area of a Triangle with vertices

Area

Also Area of

Given

Question 4: The points and are the vertices of a quadrilateral . Determine whether is a rhombus or not.

Answer:

Given and

Since , therefore is not a rhombus.

Question 5: Find the coordinates of the center of the circle inscribed in a triangle whose vertices are and .

Answer:

Given the vertices of the triangle and .

We know that if the vertices of a triangle are , then the in-center of the circle is

where

Now

Therefore the coordinates of the in-center are

Question 6: The base of an equilateral triangle with side lies along the y-axis such that the mid-point of the base is at the origin. Find the vertices of the triangle

Answer:

Given is an equilateral triangle with side . Let be .

Therefore the vertices are or

Question 7: Find the distance between and when (i) is parallel to the y-axis (ii) is parallel to the x-axis.

Answer:

Given and

We know

i) When is parallel to x-axis, then

i) When is parallel to y-axis, then

Question 8: Find a point on the x-axis, which is equidistant from the point and .

Answer:

Given points and

Let be the equidistant point from and