Note: Relation between the areas of two similar triangles: If then
Question 1: (i) The ratio between the corresponding sides of two similar triangles is . Find the ratios between the areas of these triangles.
(ii) Areas of two similar triangles is and
. Find the ratios between the length of their corresponding sides.
Answer:
(i) Required ratio of their areas
(ii)
Therefore Required ratio
Question 2: A line is drawn parallel to the base
which meets sides
at points
respectively. If
; find the value of
(i)
(ii)
Answer:
Given
Consider
(alternate angles)
(alternate angles)
Therefore (AAA postulate)
Hence
Also
Question 3: The perimeter of two similar triangles are and
. If one side of the first triangle is
, determine the corresponding side of the second triangle.
Answer:
Since the two given triangles are similar, we have
Question 4: In the given figure,
. Find:
(i) the length of , if the length of
is
.
(ii) the ratio between the areas of trapezium and
.
Answer:
Given
Consider
(alternate angles)
(common angle)
Therefore
Therefore
Also
Question 5: is a triangle.
is a line segment intersecting
and
such that
and divides
into two parts equal in area. Find the value of ratio
.
Answer:
Given
Consider
(corresponding angles)
(common angle)
Therefore
Therefore
Question 6: In the given
and
. Calculate the value of the ratio:
(i) and then
(ii)
(iii)
Answer:
(i) Given and
.
Consider
(alternate angles)
(common angle)
Therefore
Therefore
(ii) Since have common vertex
and their bases
are along the same straight line
Consider
(alternate angles)
(common angle)
Therefore
Therefore
(iii) Since have common vertex
and their bases
are along the same straight line
Therefore
Question 7: The given diagram shows two isosceles triangles which are similar.
are not parallel.
and
. Calculate:
(i) the length of
(ii) the ratio of the areas of
Answer:
Given
Since
Solving
(ii)
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: In the given figure,
. Area of
, area of trapezium
and
. Calculate the length of
. Also find the area of
.
Answer:
Given
Consider
(alternate angles)
(common angle)
Therefore
Now Area of trapezium
Area of
Question 10: The given figure shows a trapezium in which
and diagonals
intersect at point
. If
. Find:
(i)
(ii)
(iii)
(iv)
Answer:
(i) Since have common vertex
and their bases
are along the same straight line
Therefore
(ii) Since
Therefore
(iii) Since have common vertex
and their bases
are along the same straight line
Therefore
(iv) Since have common vertex
and their bases
are along the same straight line
Therefore