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

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