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 $latex\displaystyle DE = \frac{1}{4} BC $.
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