Question 1: If the line segment joining the points and subtends an angle at the origin , prove that: .
Consider the figure given.
Using cosine formula in we get
Question 2: The vertices of a and and . Find
We know where
Question 3: Four points ) and are given in such a way that , find .
Given ) and
We know that the area of a Triangle with vertices
Also Area of
Question 4: The points and are the vertices of a quadrilateral . Determine whether is a rhombus or not.
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 .
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
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
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.
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 .
Given points and
Let be the equidistant point from and