*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

Question 11: On a map drawn to a scale of a triangular plot of of land has the following measurements: . Calculate:

(i) the actual length of in kilometers

(ii) the actual area of the plot in

Answer:

Scale factor

(i) Length of side in the map = the actual length of the side in the land

Length of side in the map the actual length of the side in the land

actual length of

Hence

(ii) Area of the plot

Question 12: 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 13: 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 14: In the given figure , . . Find the ratio between the area of the .

Answer:

Consider

(Since is common and is given)

Therefore

We know that for similar triangles

Question 15: is an isosceles triangle in which and . If , find:

(i)

(ii)

Answer:

(i) Consider

(Since angles opposite equal sides are also equal)

Therefore

We know that for similar triangles

(ii) Now Consider

(given)

Therefore

Therefore

Question 16: An airplane is long and its model is long. If the total outer surface area of the model is , find the cost of painting the outer surface of the airplane at the rate of . Given that of the surface of the airplane sis left for windows.

Answer:

of the model represent of actual airplane

Therefore wold represent of actual airplane

Hence would represent of the surface are of the actual airplane

Given that the surface area of the model .

Therefore the actual surface are of the actual airplane

Area to be painted

Cost of painting

Therefore the total cost of painting