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.