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