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