Note: We are going to use the following formula extensively in solving the following problems. The Distance between any two points and is
Notes: If a point is on , its ordinate is , therefore the point on is taken as . Similarly, if the point is on , its abscissa is , therefore the point on is taken as . For details refer to the following lecture notes.
Question 1: Find the distance between the following pairs of points:
i) and
ii) and
iii) and
iv) and
Answer:
i) and
Distance
ii) and
Distance
iii) and (
Distance
iv) and
Distance
Question 2: Find the distance between the origin and the points:
i)
ii)
iii)
Answer:
i)
Distance
ii)
Distance
iii)
Distance
Question 3: The distance between point and is . Find .
Answer:
Distance:
Question 4: Find the coordinate of the point on which are at a distance of units from the point .
Answer:
Distance:
Therefore the points are
Question 5: Find the coordinate of the point on which are at a distance of units from the point .
Answer:
Distance:
Hence the points could be
Question 6: A point is at a distance of units from the point . Find the coordinates of the point if its ordinate is twice its abscissa.
Answer:
Distance:
Hence the points could be
Question 7: A point is equidistant from the point and . Find .
Answer:
is equidistant from and
Therefore
Question 8: What point on is equidistant from the point and .
Answer:
Let the point be . Therefore
Therefore the point is
Question 9: What point on is equidistant from the point and .
Answer:
Let the point be . Therefore
Hence the point is
Question 10: A point lies on and another point lies on . Write the ordinate of point , abscissa of point . If the abscissa of point is and ordinate of point is . Calculate the length of the line segment .
Answer:
Let and . Given and
Distance:
Question 11: Show that the points and are the vertices of an isosceles triangle.
Answer:
and
Therefore two sides are equal which makes it an isosceles triangle.
Question 12: Prove that the points and are the vertices of the rectangle .
Answer:
and
Therefore and .
Hence it is a rectangle.
Question 13: Prove that the points and are the vertices of an isosceles triangle. Find the area of the triangle.
Answer:
and
For this to be a right angled triangle we should have
. Hence proved that it is a right angled triangle.
Area sq. units.
Question 14: Show that the points and are the vertices of the square .
Answer:
and
Therefore .
Hence it is a square.
Question 15: Show that and are the vertices of a rhombus.
Answer:
and
Therefore .
Two sides are equal and the other two sides are of different length. Hence it is a rhombus.
Question 16: Points and are the vertices of a quadrilateral . Find a if a is negative and .
Answer:
and
Therefore
. Hence as it is negative.
Question 17: The vertices of a triangle are and . Find the coordinates of the circumcenter of the triangle.
Answer:
and are the points
Let the coordinates of the circumcenter
Therefore
Hence
Therefore equation 1:
Also equation 2:
Hence the coordinates of the circumcenter is
Question 18: Given and . Find if .
Answer:
and
Therefore
Question 19: Given and . Find if .
Answer:
and
Question 20: The center of the circle is . Find if the circle passes through and the length of the diameter is units.
Answer:
Diameter units i.e.Radius units
Question 21: The length of the line is units and the coordinates of are , calculate the coordinates of point , if its abscissca is .
Answer:
and Let
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