*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