Question 1: In the adjoining figure, compute the area of quadrilateral .
Answer:
Therefore
Therefore
Question 2: In the adjoining figure, is a square and
and
are, respectively, the mid points of
and
. Find the area of
if
.
Answer:
is a square
Since are mid points of
respectively,
Similarly,
Question 3: Compute the area of trapezium in the adjoining figure.
Answer:
Question 4: In the adjoining figure, and
. Find the area of
.
Answer:
Since mid point of hypotenuse is equidistant from all three vertices
Question 5: In the adjoining figure, is a trapezium in which
and the distance between
and
is
. Find the value of
and area of trapezium
.
Answer:
Question 6: In the adjoining figure, is a rectangle inscribed in a quadrant of a circle of radius
. If
, find the area of the rectangle.
Answer:
is a rectangle.
Question 7: In the adjoining figure, is a trapezium in which
. Prove that
Answer:
Given
Since and
are between the same parallels and have the same base, therefore
Question 8: In the adjoining figure, and
are parallelograms. Prove that
Answer:
is a parallelogram
Similarly, is a parallelogram
Since
(By SSS criterion)
Hence
Question 9: Diagonals and
of a quadrilateral
intersect each other at
. Show that
Answer:
Question 10: In the adjoining figure, and
are two triangles on the base
. If the line segment
bisected by
at
, show that
Answer:
Now consider and
(given)
(Vertically opposite angles)
(altitudes)
Question 11: If is any point in the interior of a parallelogram
, then prove that the area of the
is less than half the area of the parallelogram.
Answer:
We know
Question 12: If is the median of
, then prove that
and
are equal in area. If
is the mid point of median
, prove that
Answer:
Since
Since
Now
Since
Question 13: A point is taken on the side of
and of a
such that
. Prove that
Answer:
Question 14: is a parallelogram whose diagonals intersect at
. If
is any point on
, prove that
i) ii)
Answer:
i) Since diagonals of a parallelogram bisect each other. Therefore is the mid point of
as well as
In is the median,
ii) In , since
is the median
In , since
is the median
Question 15: is a parallelogram in which
is produced to
such that
.
intersects
at
.
i) Prove that
ii) If the area of , find the area of
Answer:
Given
Consider and
since
( By AAS criterion)
Since is median in
Question 16: is a parallelogram whose diagonals
and
intersect at
. A line through
intersect
at
and
at
. Prove that
Answer:
Consider and
(diagonals bisect each other)
(vertically opposite angles)
(since
and
is a transversal)
Question 17: is a parallelogram.
is a point on
such that
and
is the point on
such that
. Prove that
is a parallelogram whose area is one third of the area of parallelogram
.
Answer:
is a
Also (since
)
and equal to each other.
is a
Question 18: In a and
are respectively the mid points of
and
and
is the mid point of
. Prove that:
i) ii)
iii)
Answer:
i) In is the median
… … … … … i)
In is the median
… … … … … ii)
In is the median
… … … … … iii)
From i), ii) and iii) we get
… … … … … iv)
Similarly, … … … … … v)
From iv) and v) we get
ii) In is the median
In is the median
From (i) we have
and
iii) Since is the median of
Since is the median of
Question 19: is a parallelogram,
is the point on
such that
is a point of
such that
and
is a point of
such that
. Prove that
i) ii)
iii)
Answer:
i) Given is a
ii)
iii) In , construct altitude
from
on
and
Question 20: In the adjoining figure, and
.
i) Name a triangle equal in area of
ii) Prove that
iii) Prove that
Answer:
Given and
i) Consider
and
have the same base are between the same parallels.
ii) and
are between the same parallels
iii) From i)
From ii)
Question 21: In the adjoining area, is a parallelogram in which
and
. Prove that
Answer:
is a
Also
Since
and
Question 22: In the adjoining figure, is a trapezium in which
and
and
. If
and
are, respectively the mind points of
and
, prove that i)
ii)
is a trapezium iii)
Answer:
i) Construction: Join and extend it meeting
at
Consider and
(given)
(vertically opposite angles)
(since
) (Alternate angles)
and
Since and
are mid points of
and
respectively,
ii) From i) we have and
(Given)
is a trapezium
iii) Since and
are mid points of
and
respectively, and
, the distance between
and
and
and
are the same.
Let the distance between and
be
Therefore
Question 23: In the adjoining figure, and
are two equilateral triangles such that
is the mid point of
.
intersects
at
. Prove that
i)
ii)
iii)
Answer:
i) and
are equilateral triangles (given)
(given)
Altitude of
Altitude of
ii) Since (alternate angles)
Since is median in
Since and
are between the same parallels and have the same base
iii) Since (alternate angles)
Question 24: is the mid point of side
of
and
is the mid point of
. If
is the mid point of
, prove that
Answer:
In , since
is the median
… … … … … i)
In , since
is the median
… … … … … ii)
In , since
is the median
… … … … … iii)
Question 25: In the adjoining figure, and
are mid points of
and
respectively,
and
and
are straight lines. Prove that
Answer:
In is the mid point of
and
is the mid point of $latex AB
Since also
Also (Between the same parallels)
Similarly,
Therefore since the bases are equal and triangles are between the same parallels,
Question 26: In the adjoining figure, and
are two parallelogram. Prove that
i) ii)
iii)
Answer:
i) Consider and
(corresponding angles)
(opposite sides)
(corresponding angles)
ii) From i) … … … … … i)
… … … … … ii)
(Same base and between same two parallels)
Dividing i) by ii)
iii) From i) since
Question 27: In the adjoining figure, is a
.
is any point on
.
and
. Prove that
Answer:
Since and
Since and
Similarly,
Since and
from i) and ii) and iii) we get
Question 28: In a , if
and
are point on
and
respectively such that
. Prove that:
i) ii)
iii) iv)
Answer:
i) Since and
are on the same base and between two parallels,
ii) Similarly,
Base is the same and between same parallels
iii)
iv)
Question 29: In the adjoining figure, and
are two points on
such that
. Show that
Answer:
All the three triangles have equal bases and are between the same parallels.
Question 30: In the adjoining figure, is a right triangle right angled at
and
are squares on the sides of
and
respectively. Line segment
meets
at
. Show that:
i) ii)
iii) iv)
v) vi)
vii)
Answer:
i) Consider and
(since )
… … … … … i) (By SAS criterion)
ii) Consider and
Since and
are on the same base an
and between the same parallels
from i)
iii) Consider
iv) Consider and
(By SAS criterion)
v)
… … … … … iv)
vi) and rectangle
are having the same base
and are between the same parallels
… … … … … v)
From iv) and v) we get
v)
Question 31: In the adjoining figure, and
are two parallelograms of equal area. Prove that
is parallel to
.
Answer:
Given
Since the two triangles have the same base,
Question 32: Prove that the area of the quadrilateral formed by joining the mid points of the adjacent sides of a quadrilateral is half the area of the given quadrilateral.
Answer:
Construction: Join and
.
is the median in
… … … … … i)
is median in
… … … … … ii)
From i) and ii) we get
… … … … … iii)
Similarly,
… … … … … iv)
Adding iii) and iv) we get
… … … … … v)
Similarly we can prove that
… … … … … vi)
Adding v) and vi) we get
Question 33: In the adjoining figure, is a parallelogram.
is the mid point of
and
meets diagonal
at
. If area of
.
i) ii) area of
iii) area of
Answer:
i) Given is a
Consider and
(vertically opposite angles)
(alternate angles)
(alternate angles)
ii) Since bases and
of
and
lie on the same line, and have the same height,
iii)
Similarly,
Question 34: If and
are mid points of the sides
and
respectively of
, prove that
is a trapezium. Also find it’s area if area of
is
.
Answer:
Since and
are mid points of
and
respectively,
is a trapezium.
Draw altitude from .
, and since
We can prove (by AAA criterion)
Question 35: In the adjoining figure, is any point on median
of
. Prove that, i)
ii)
Answer:
i) Since is the mid point of
and
and
have the same height,
ii) Since is the median
Question 36: In the adjoining figure, if , prove that
i) ii)
Answer:
i) Given
… … … … … i)
ii) from i)
Question 37: If in a quadrilateral , diagonal
bisects the diagonal
, then prove that
Answer:
In is a median
… … … … … i)
Similarly, … … … … … ii)
Adding i) and ii)
Question 38: In the adjoining figure, is a point on the side
of
such that
and
is a pint on
such that
. Find
Answer:
Question 39: In the adjoining figure, is the median of the
and
is the point on
such that
. Find
i) ii)
Answer:
Given is median
i) Also
Given,
or
ii) from i)
Question 40: In the adjoining figure, is the midpoint of the side
of
and
is a parallelogram. If
, find
Answer:
Consider and
and
Also since
Question 41: In the adjoining figure, and
are mid points of sides
and
respectively of parallelogram
. If
find
. Name the parallelogram whose area is equal to the area of
.
Answer:
. Join
Question 42: In the adjoining figure, is a point on side
of a parallelogram
such that
. If
produced meets
produced at
and
, find
and
.
Answer:
Given,
We can prove
Also
Question 43: In the adjoining figure, and
are two parallelograms. Prove that
Answer:
Join
and
are on the same base and between the same parallel
Similarly,
Question 44: is a square.
and
are mid points of the sides
and
respectively. Prove that
Answer:
… … … … … i)
Similarly,
… … … … … ii)
Adding i) and ii)
Question 45: A point is taken on the sides
and of
and
is produced to
such that
, prove that
Answer:
Draw a line to
… … … … … i)
(They have equal bases and between the same parallels)
Similarly, … … … … … ii)
Adding i) and ii) we get
Question 46: In the adjoining figure, if and
and
. Prove that
Answer:
Consider and
(By ASA criterion)
Now consider, and
(By ASA criterion)
Question 47: is a rectangle and
is mid point of
.
is produced to meet
at
. Prove that
Answer:
Consider and
Question 48: If each diagonal of a quadrilateral divides it into two triangles of equal areas, then prove that it is a parallelogram.
Answer:
divides
into two equal halves
Similarly,
Similarly,
is a parallelogram.