Question 1: In the adjoining figure, compute the area of quadrilateral .

Answer:

Question 2: In the adjoining figure, is a square are, respectively, the mid points of . Find the area of if .

Answer:

is a square

Since are mid points of respectively,

Question 3: Compute the area of trapezium in the adjoining figure.

Answer:

Question 4: In the adjoining figure, . 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 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:

Since are between the same parallels and have the same base, therefore

Question 8: In the adjoining figure, are parallelograms. Prove that

Answer:

is a parallelogram

is a parallelogram

Since

(By SSS criterion)

Hence

Question 9: Diagonals of a quadrilateral intersect each other at . Show that

Answer:

Question 10: In the adjoining figure, are two triangles on the base . If the line segment bisected by at , show that

Answer:

Now consider

(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 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

Answer:

i) Since diagonals of a parallelogram bisect each other. 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:

Consider

since

( By AAS criterion)

Since is median in

Question 16: is a parallelogram whose diagonals intersect at . A line through intersect at at . Prove that

Answer:

Consider

(diagonals bisect each other)

(vertically opposite angles)

(since is a transversal)

Question 17: is a parallelogram. is a point on such that 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 are respectively the mid points of is the mid point of . Prove that:

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)

… … … … … v)

From iv) and v) we get

ii) In is the median

In is the median

From (i) we have

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 is a point of such that . Prove thatÂ

Â

Answer:

is a

iii) In , construct altitude from on

Question 20: In the adjoining figure, .Â

i) Name a triangle equal in area of

ii) Prove that

iii) Prove that

Answer:

i) Consider

have the same base are between the same parallels.

are between the same parallels

iii) From i)

From

Question 21: In the adjoining area, is a parallelogram in which . Prove that

Answer:

is a

Also

Since

Question 22: In the adjoining figure, is a trapezium in which . If are, respectively the mind points of , prove that is a trapezium

Answer:

i) Construction: Join and extend it meeting at

Consider

(given)

(vertically opposite angles)

(since ) (Alternate angles)

Since are mid points of respectively,Â

ii) From i) we have (Given)

is a trapezium

iii) Since are mid points of respectively, , the distance between are the same.

Let the distance between be

Â

Question 23: In the adjoining figure, are two equilateral triangles such that is the mid point of . intersects at . Prove that

Answer:

are equilateral triangles (given)

(given)

Altitude of

Altitude of

ii) Since (alternate angles)

Since is median in

Since are between the same parallels and have the same base

iii) Since (alternate angles)

Question 24: is the mid point of side of 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, are mid points of respectively, are straight lines. Prove that

Answer:

In is the mid point of is the mid point of $latex \displaystyle AB

Since also

Also (Between the same parallels)

Therefore since the bases are equal and triangles are between the same parallels,Â

Question 26: In the adjoining figure, are two parallelogram. Prove that

Answer:

i) Consider

(corresponding angles)

(opposite sides)

(corresponding angles)

ii) From … … … … … i)

… … … … … ii)

(Same base and between same two parallels)

Dividing i) by ii)

Â

iii) From i) since

Question 27: In the adjoining figure, is a . is any point on . . Prove that

Answer:

Since

Since

Since

from i) and ii) and iii) we get

Question 28: In a , if are point on respectively such that . Prove that:

Answer:

i) Since are on the same base and between two parallels,

Base is the same and between same parallels

Question 29: In the adjoining figure, 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 are squares on the sides of respectively. Line segment meets at . Show that:

v

v

Answer:

i) Consider

(since )

… … … … … i) (By SAS criterion)

ii) Consider

Since are on the same base an and between the same parallels

from i)

iii) Consider

iv) Consider

(By SAS criterion)

… … … … … iv)

and rectangle are having the same base and are between the same parallels

… … … … … v)

From iv) and v) we get

Question 31: In the adjoining figure, are two parallelograms of equal area. Prove that is parallel to .

Answer:

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 .

is the median in

… … … … … i)

is median in

… … … … … ii)

From i) and ii) we get

… … … … … iii)

… … … … … 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 meets diagonal at . If area of .

ii) area of iii) area of

Answer:

is a

Consider

(vertically opposite angles)

(alternate angles)

(alternate angles)

Â

Â

ii) Since bases of lie on the same line, and have the same height,

Question 34: If are mid points of the sides respectively of , prove that is a trapezium. Also find it’s area if area of is .

Answer:

Since are mid points of respectively,

is a trapezium.

Draw altitude from .

, and since

We can prove (by AAA criterion)

Â

Question 35: In the adjoining figure, is any point on median of . Prove that,

Answer:

i) Since is the mid point of have the same height,

ii) Since is the median

Question 36: In the adjoining figure, if , prove that

Answer:

… … … … … i)

ii) from

Question 37: If in a quadrilateral , diagonal bisects the diagonal , then prove that Â

Answer:

In is a median

… … … … … i)

… … … … … ii)

Adding i)

Question 38: In the adjoining figure, is a point on the side of such that is a pint on such that . Find

Answer:

Question 39: In the adjoining figure, is the median of the is the point on such that . Find

Answer:

is median

Â

i) Also

Given,

Â

Â

Â

Â

or Â

ii) from Â

Â

Question 40: In the adjoining figure, is the midpoint of the side of is a parallelogram. If , find

Answer:

Consider

Also since

Question 41: In the adjoining figure, are mid points of sides 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 , find .

Answer:

Given,

We can prove

Also

Question 43: In the adjoining figure, are two parallelograms. Prove that

Answer:

Join

are on the same base and between the same parallel

Question 44: is a square. are mid points of the sides respectively. Prove that

Answer:

… … … … … i)

… … … … … ii)

Adding i) and ii)

Question 45: A point is taken on the sides and of is produced to such that , prove that

Answer:

Draw a line to

… … … … … i)

(They have equal bases and between the same parallels)

… … … … … ii)

Adding i) and ii) we get

Question 46: In the adjoining figure, if . Prove that

Answer:

Consider

(By ASA criterion)

Now consider,

(By ASA criterion)

Question 47: is a rectangle is mid point of . is produced to meet at . Prove that

Answer:

Consider

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

is a parallelogram.