Question 1: In the given figure,
and
are perpendicular to
. If
, calculate
. [2005]
Answer:
In and
:
(perpendiculars)
(common angle)
(by AAA postulate)
Therefore
Question 2: In the given figure,
and
are right angled at
and
respectively. Given
and
. (i) Prove
(ii) Find
and
. [2012]
Answer:
In
(common angle)
(right angles)
Therefore (AAA postulate)
Since
Given AC=10 cm, AP = 15 cm and PM= 12 cm
Question 3: 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 4: 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 5: In the right angled
is the altitude. Given that
and
, calculate the value of
. [2000]
Answer:
(common)
(AAA postulate)
Question 6: In the given figure
.
(i) Prove that and
are similar.
(ii) Given that
, calculate
, if
. Also find:
and . [2004]
Answer:
(i)
(corresponding angles)
and (corresponding angles)
Therefore (AAA postulate)
(ii) Since
Given
(iii) Since
Question 7: In the figure given below,
and
are perpendiculars to the line segment
. If
and area of
, find the area of
. [2006]
Answer:
In
(vertically opposite angles)
Therefore (By AAA postulate)
Question 8: In the figure given below,
is a parallelogram.
is a point on
such that
.
produced meets
produced at
. Given the area of
. Calculate:
(i) area of
(ii) area of parallelogram . [1996]
Answer:
(i) In
(vertically opposite angles)
(alternate angles)
Therefore
Also
(ii) In
(Given)
(corresponding angles are equal)
(common angle)
Therefore
Also
Question 9: A model of a ship is made to scale of .
(i) The length of the model is ; calculate the length of the ship.
(ii) The area of the deck of the ship is ; find the area of the deck of the model.
(iii) The volume of the model is ; calculate the volume of the ship in
. [1995]
Answer:
Scale factor
(i) Length of the model Actual length of the ship
Actual length of the ship
(ii) Area of the deck of the model area of the deck of the actual ship
(iii) Volume of the model Volume of the actual ship
Question 10: In the figure given below
is a triangle.
is parallel to
and
.
(i) Determine the ratios
(ii) Prove that is similar to
. Hence , find
. [2007]
Answer:
(i) Given
(common angle)
(ii) In
(alternate angles)
(vertically opposite angles)
(iii) We know
Question 11: In
.
(i) Prove that is similar to
.
(ii) Find .
(iii) Find . [2014]
Answer:
(i) In
(common angle)
(given)
(AAA postulate)
(ii) Since
(iii) Since
Therefore
Hence
Question 12: In the following figure, are parallel lines.
. Calculate:
. [1985]
Answer:
Consider
(Corresponding angles)
(common angle)
(AAA Postulate)
Therefore
Now consider
(Vertically opposite angles)
(Alternate angles)
(AAA postulate)
Therefore
Also
Question 13: In
and
are two points on the base
, such that
and
. Prove that:
(i)
(ii)
(iii) . [2003]
Answer:
(i) Consider
(Given)
(Given)
(AAA postulate)
(ii) Since
(iii) Consider
(Given)
(common angle)
(AAA postulate)
Therefore
Question 14: In the given figure,
is a triangle with
. Prove that
. If
and area of
. Calculate the:
(i) length of
(ii) area of [2010]
Answer:
Consider
(given)
(common)
Therefore
(AAA postulate)
(i) Given
(ii)
Area of
Question 15: In the given figure
is a right angled triangle with
.
(i) Prove
(ii) If , find
(iii) Find the ratio of the area of is to area of
. [2011]
Answer:
(i) Let
Therefore
Therefore (AAA postulate)
(ii)
Therefore
(iii)
Question 16: In the given figure
and
are perpendiculars to
.
(i) Prove that
(ii) If and
, calculate
(iii) Find the ratio of the . [2013]
Answer:
(i) From
(given)
(common angle)
(AAA postulate)
(ii) Since
In ,
and
Therefore
(iii) Since
Therefore the