Question 1: State True or False:
(i) Two similar polygons are necessarily congruent – False
(ii) Two congruent polygons are necessarily similar – True
(iii) All equi-angular triangles are similar – True
(iv) All isosceles triangles are similar – False
(v) Two isosceles right angles triangles are similar – True
(vi) Two isosceles triangles are similar, if an angle of one is congruent to the corresponding angle of the other – True
(vii) The diagonals of the trapezium, divide each into proportional segments – True
Question 2: In , where
are points on
respectively. Prove that
. Also find the length of
.
Answer:
In
Consider
(alternate angles)
(alternate angles)
Therefore (AAA postulate)
Hence
Question 3: Given
,
and
. Find the length of the segments
Answer:
Consider
(Given)
Therefore (AAA postulate)
Substituting
Question 4: is a point of side
of
such that
. Prove that
.
Answer:
Consider
(Given)
Therefore (AAA postulate)
Therefore
Question 5: In the given figure,
and
are right angled at
and
respectively. Given
and
. (i) Prove
(ii) Find
and
. [2012]
Answer:
In
(common angle)
Therefore (AAA postulate)
Since
Given AC=10 cm, AP = 15 cm and PM= 12 cm
Question 6: are points in sides
respectively of parallelogram
. If diagonal
and segment
intersect at
; prove that:
.
Answer:
In
(opposite angles)
(alternate angles)
Therefore (AAA postulate)
Therefore
Question 7: Given are altitudes of
. Prove that (i)
(ii)
Answer:
(i) Consider
(Given, altitudes)
Therefore (AAA postulate)
(ii)
Question 8: Given
is a rhombus,
are straight lines. Prove that
.
Answer:
In
(vertically opposite angles)
(alternate angles)
Therefore (AAA postulate)
Therefore
(
is a Rhombus)
Question 9: Given . Prove
Answer:
and
.
Therefore
Consider
(Given)
Therefore (AAA postulate)
Question 10: In and the bisector of
meets
at point
. Prove that (i)
(ii)
Answer:
Given
Therefore
Consider
(Given)
Therefore (AAA postulate)
(
)
Question 11: In and
. Prove that:
(i)
(ii)
(iii)
Answer:
(i) Given and
Consider
(given)
(common angle)
Therefore (AAA postulate)
Hence
Therefore
(ii) Consider
(given)
(common angle)
Therefore (AAA postulate)
Hence
Therefore
(iii) Adding (i) and (ii)
Question 12: In . Prove that
.
Answer:
.
Consider
(given)
(common angle)
Therefore (AAA postulate)
Hence
Therefore
Question 13: In .
(i) If
(ii) If
(iii) If
Answer:
Consider
(given)
Therefore (AAA postulate)
(i)
Therefore
(ii)
Therefore
(iii)
Therefore
Question 14: In the figure,
is a parallelogram with
and
.
is a point on
such that
.
produced meets
at
and
produced at
. Find the lengths of
and
. [1997]
Answer:
In and
(vertically opposite angles)
(alternate angles)
Therefore (AAA postulate)
Therefore
In
(vertically opposite angles)
(alternate angles)
Therefore (AAA postulate)
Therefore
Question 15: In quadrilateral , Diagonal
intersect at point
such that:
. Show that
is a parallelogram.
Answer:
Given
… … … … (i)
In , because
, by proportionality theorem
… … … … (i)
Therefore from (i) and (ii)
Similarly in
Also it is given that
Therefore . Therefore
is a parallelogram.
Question 16: Given
and
. Prove that:
(i)
(ii)
Answer:
(i) Given and
In
… … … … (i)
In
… … … … (ii)
From (i) and (ii) you get
(ii) Consider
We have
and (common angle)
Therefore
Question 17: In and
. Show that
.
Answer:
Given
Also given
Therefore (SAS postulate)
Therefore
Adding
… … … (i)
We know … … … … (ii)
From (i) and (ii)
Question 18: In the given figure
;
and
.
(i) Name the three pairs of similar triangles
(ii) Find the lengths of
Answer:
Given , The three pairs of similar triangles are
Since
Since
Question 19: In the given figure
, and
. Prove that
.
Answer:
Given , and
Using basic proportionality theorem
In
… … … … (i)
In
… … … … (ii)
From (i) and (ii) we get
Question 20: Through the mid point of the side
of a parallelogram
, the line
is drawn intersecting diagonal
and
produced in
. Prove that
.
Answer:
Consider
(alternate angles)
(opposite angles)
(as M is the midpoint of CD)
Therefore
Therefore (corresponding angles)
(opposite sides of a parallelogram)
Therefore
Now consider
Therefore
Hence
Hence
Question 21: In the figure given below,
is a point on
such that
.
.
(i) Calculate the ratio , giving reasons for your answer.
(ii) In and in
. Given
, calculate length of
. [1999]
Answer:
(i) Given
Also , applying basic proportionality theorem
(corresponding angles)
(corresponding angles)
Therefore (AAA postulate)
(ii) Given
(alternate angles)
Therefore (AAA postulate)
Question 22: In the right angled
is the altitude. Given that
and
, calculate the value of
. [2000]
Answer:
(common)
(AAA postulate)
Question 23: In the figure given below, median of the
meet at
. Prove that:
(i)
(ii) from (i)
Answer:
Given are medians
Therefore and
Applying converse of proportionality theorem
In and
(alternate angles)
(vertically opposite angles)
Therefore (AAA postulate)
Therefore
In and
(corresponding angles)
(common angle)
Therefore (AAA postulate)
Therefore
From 1,