Question 1: In a and are, respectively, the mid-points of and . If the lengths of side and are and , respectively, find the perimeter of .

Answer:

Given: and

Since and are mid points of and respectively,

Similarly

and

Therefore perimeter of

Question 2: In a triangle and . Find the measures of the angles of the triangle formed by joining the mid-points of the sides of this triangle.

Answer:

We know,

Therefore

Question 3: In a triangle, and are the mid-points of sides and respectively. If and , find the perimeter of the quadrilateral .

Answer:

Therefore perimeter of

Question 4: In a median is produced to such that . Prove that is a parallelogram.

Answer:

Given and

Since the diagonals of a parallelogram bisects each other we can say that is a parallelogram.

Question 5: In a and are the mid-points of and respectively. The altitude to intersects at . Prove that .

Answer:

Given (since is mid point of )

(since is the mid point of )

Since and are mid point,

Since and is the mid point of is the mid point of

Therefore

Question 6: In a and are perpendiculars from and respectively on any line passing through . If is the mid-point of , prove that .

Answer:

Consider the diagram shown.

Construction :- Draw

Now, If a transversal makes equal intercepts on three or more parallel lines, then any other transversal intersecting them will also make equal intercepts.

and are perpendicular to Therefore,

Now, and is the transversal making equal intercepts i.e. .

Therefore, transversal will also make equal intercepts

Consider and ,

is common

Therefore (By S.A.S criterion)

Therefore

Question 7: In the adjoining figure, is right-angled at . Given that and are the mid-points of the sides and respectively, calculate i) The length of ii) The area of .

Answer:

i)

ii) We know:

Therefore

Therefore Area of

Question 8: In adjoining figure, and are the mid-points of and respectively. If and , calculate and .

Answer:

Given and and and are the mid-points of and respectively

Question 9: is a triangle and through lines are drawn parallel to and respectively intersecting at and . Prove that the perimeter of is double the perimeter of .

Answer:

To prove:

Consider, and . Both are parallelograms.

Hence and

Similarly, we can prove that is mid point of and is mid point of .

Hence and

Therefore

Question 10: In the adjoining figure, . is any line from to intersecting in and are respectively the mid-points of and . Prove that

Answer:

Since are midpoints of and respectively,

In

Similarly, Since are midpoints of and respectively,

In

Given

Therefore

Question 11: In adjoining figure, and and is the bisector of exterior of . Prove that (i) (ii) is a parallelogram.

Answer:

i)

Therefore

Since

Hence

ii) If were to be to then

Therefore

Already given that . Hence is a parallelogram.

Question 12: is a kite having and . Prove that the figure formed by joining the mid-points of the sides, in order, is a rectangle.

Answer:

Given:

Construction: Join

To Prove: and

Also

Proof: Consider

Since and are midpoints of and respectively,

Similarly, Since and are midpoints of and respectively,

Using the same logic,

Since and are midpoints of and respectively,

Similarly, Since and are midpoints of and respectively,

Also

Similarly,

Now consider and

(given)

(diagonals bisect each other)

is common

Therefore

Similarly,

Hence we can say that is a rectangle because opposites sides are equal and parallel and the sides are perpendicular.

Question 13: Let ABC be an isosceles triangle in which . lf be the mid-points of the sides and respectively, show that the segment and, bisect each other at right angles.

Answer:

Given:

Since is the midpoint of , therefore

Similarly, is the midpoint of , therefore

Also

is also midpoint of , therefore

(given)

Therefore

Since is isosceles triangle,

Because

Consider and

is common

Therefore (By S.A.S criterion)

Now,

Question 14: is a triangle. is a point on such that and is a point on such that . Prove that .

Answer:

Let and be midpoints of and respectively.

Therefore

In

Question 15: In the adjoining figure, is a parallelogram in which is the mid-point of and is a point on such that . If produced meets at , prove that is a mid-point of .

Answer:

To prove:

Given:

Let

Therefore is midpoint of

In and are midpoint of and respectively.

Therefore

Since, is the midpoint of and

is the midpoint of

Question 16: In adjoining figure, and are rectangles and is the mid-point of . Prove that (i) (ii)

Answer:

i) Since is the midpoint of and is the midpoint of

ii) Similarly, is the midpoint of and is the midpoint of

(since diagonals of a rectangle are equal)

Question 17: is a parallelogram, and are the mid-points of and respectively. is any line intersecting and at and respectively. Prove that .

Answer:

Since is the midpoint of and is the midpoint of

And

But

Since

is a parallelogram

Since is the midpoint of ,

Therefore

Question 18: Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

Answer:

Similarly,

Therefore is a parallelogram

bisect each other as diagonals of a parallelogram bisect each other.

Question 19: Fill in the blanks to make the following statements correct:

(i) The triangle formed by joining the mid-points of the sides of an isosceles triangle is Isosceles Triangle

(ii) The triangle formed by joining the mid-points of the sides of a right triangle is Right Angled

(iii) The figure formed by joining the mid-points of consecutive sides of a quadrilateral is Parallelogram