If there are two points
Distance between two points
This is known as distance formula in the world of Coordinate Geometry.
It has been derived from the famous Pythagorean theorem. All problems in this page are related to
finding distance between two points.
To find the distance of a point, say
Distance between origin and one point
which is the simplified form of
Q. 1(i): Find the distance between the pairs of points
Solution
For points
Q. 1(ii): Find the distance between the pairs of points
Solution
For points
Q. 1(iii) Find the distance between the pairs of points
Solution
Here,
Q.2. Find the distance between the pairs of points
Solution
Here,
Q.3. Determine if the points
Solution
Let us name the given points as
For PQ,
Following is the graph of the three points P, Q and R.
Q. 4.Check whether
Solution
Let us name the given points as
We calculate the distances between
P and Q, Q and R, P and R.
For PQ,
Since side PQ = side QR, PQR is an isosceles triangle.
So, points
An isosceles triangle has two equal sides.
Following is the graph of the 3 lines formed
by the given coordinates.
It is clear that
Q.5. In a classroom, 4 friends are seated at the points A, B, C and D as shown in the given figure. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, "Don’t you think ABCD is a square?" Chameli disagrees. Using distance formula, find which of them is correct.
Solution
According to the box, let us name the given points as A, B, C and D.
So the coordinates are
So, this quadrilateral is either a square or a rhombus.
Note: The diagonals of a square are always equal to each other.
But diagonals of a rhombus are unequal.
To be sure, let us compare the lengths of diagonals AC and BD ...
Q.6(i). Name the type of quadrilateral formed, if any,
by the following points, and give reasons for your answer:
Solution
Let us name the given points as
Let us measure the diagonals ...
Please check the following graph ...
Q.6(ii). Name the type of quadrilateral formed, if any,
by the following points, and give reasons for your answer:
Solution
Let us name the given points as
points
So no quadrilateral is possible with the given four points.
Ans.
Q.6(iii). Name the type of quadrilateral formed, if any,
by the following points, and give reasons for your answer:
Solution
Let us name the given points as
For distance A to B,
the opposite sides of this quadrilateral are equal.
The quadrilateral is either a rectangle or a parallelogram.
Note: The diagonals of a rectangle are always equal to each other. But diagonals of a parallelogram are unequal.
In our case, the diagonals are not equal;
So, the given quadrilateral is a parallelogram. Ans.
Q.7. Find the point on the x-axis which is equidistant
from
Solution
Given that an unkonwn point is equidistant from
It is also given that the said point lies on the X-axis.
So, from the 2nd condition we know that y coordinate of the
unknown point is 0.
Let the unknown point be
Distance between
the required point is
Q.8. Find the values of y for which the distance between the points
Solution
Distance between
Following is the graph of the two points:
Q.9. If
Solution
For distance between
For distance between
Given that distance
Distance
Distance
Following is the graph of the given 3 points with
additional 1 point:
Q.10. Find a relation between x and y such that the
point
Solution
Distance
In the diagram, points A, B and C are equidistant from points P and Q.
Their coordinates
--- x ---
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