Question 1: 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

Answer:

Question 2: In the given triangle are mid points of sides respectively. Prove that .

Answer:

In

Consider

(alternate angles)

alternate)

Therefore (AAA postulate)

Since is the mid point of

Similarly, we can prove and

Therefore

Therefore (SSS postulate)

Question 3: In the following figure are medians of . Prove that :

(i)

(ii)

Answer:

(i) Consider

(alternate angles)

(alternate angles)

Therefore (AAA postulate)

(ii) Consider

(alternate angles)

(alternate angles)

Therefore (AAA postulate)

By basic proportionality theorem

Given

Question 4: In the given figure . are altitudes where as are medians. Prove:

Answer:

(Since

Given are medians

Therefore and

… … … … (i)

Consider

Since

(alternate angles)

Therefore (AAA postulate)

… … … … (ii)

From (i) and (ii) we get

and

Therefore

Hence

Question 5: Two similar triangles are equal in area. Prove that the triangles are congruent.

Answer:

Let the two triangles be

Since the two triangles are similar, We know

Since the are of the two triangles is equal

Therefore

Question 6: The ratio between the altitudes of two similar triangles is . Write the ratios between their (i) medians (ii) perimeters (iii) areas.

Answer:

The ratio of the altitude of two similar triangles is the same as the ratio of their sides. Given ratio

(i) Ratio between their median

(ii) Ratio between their perimeter

(iii) Ratio between their areas

Question 7: The ratio between the altitudes of two similar triangles is . Find the ration between their: (i) perimeters (ii) altitudes (iii) medians

Answer:

The ratio between the altitudes of two similar triangles is .

This means that the ratio of the sides of the triangles = 4:5

(i) Ratio between their perimeter

(ii) Ratio between their altitude

(iii) Ratio between their median

Question 8: The following figure shows a in which . If and , find the length of .

Answer:

Given and

Consider

(alternate angles)

(alternate angles)

Therefore (AAA postulate)

Question 9: 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 10: On a map, drawn to a scale of , a rectangular plot of land has . Calculate:

(i) the diagonal distance of the plot in km

(ii) the area of the plot in

Answer:

Length of AB on map actual length of AB

Actual length of

Similarly Actual length of

(i) Therefore the diagonal

(ii) Area of the plot

Question 11: The dimension of a model of a multi storied building are by by . If the scale factor is , find the actual dimensions of the building. Also find:

(i) the floor area of a room of the building, if the floor are of the corresponding room in the model is

(ii) the space inside the room of them model if the space inside the corresponding room if the building is

Answer:

Dimension of model

Scale factor

Actual length

Actual breadth

Actual height

Therefore the actual dimension of the building

(i) Floor area

(ii) Volume of the model

Question 12: 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 13: In and . Prove that: .

Answer:

Consider

(Common)

(Given)

(AAA postulate)

… … … … (ii)

Consider

(Common)

(Given)

(AAA postulate)

… … … … (ii)

From (i) and (ii)

Hence Proved.

Question 14: A with is enlarged to a such that the longest side of . Find the scale factor and hence, the lengths of the other sides of .

Answer:

Scale factor

Therefore

Question 15: Two isosceles triangles have equal vertical angles. Show that the triangles are similar. If the ratios between the areas of these two triangles is , find the ratio between their corresponding altitudes.

Answer:

Consider

(Common)

(Given)

(SAS postulate)

Question 16: In and is extended to so that . Find:

(i)

(ii)

Answer:

(i) Consider

(alternate angles)

(alternate angles)

(AAA postulate)

or

(ii) Consider

(vertically opposite angles)

(alternate angles)

(AAA postulate)

Question 17:The following figure shows a in which and . Show that:

(i)

(ii)

(iii)

(iv)

Answer:

(i) Consider

(alternate angles)

(common angle)

(AAA postulate)

(ii) Therefore

(iii) Consider

(common angle)

(SAS postulate)

(iv) Therefore

Question 18: 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 19: 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 20: 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