6. Triangles

6.1 Introduction

6.2 Similar Figures

Exercise 6.1

1. Fill in the blanks using the correct word given in brackets :

(i) All circles are . (congruent, similar)

(ii) All squares are . (similar, congruent)

(iii) All triangles are similar. (isosceles, equilateral)

(iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are and (b) their corresponding sides are . (equal, proportional)

2. Give two different examples of pair of

(i) similar figures. (ii) non-similar figures.

3. State whether the following quadrilaterals are similar or not:

6.3 Similarity of Triangles

Theorem 6.1

Look at the graphs in Fig. 2.9 given below. Each is the graph of y = p(x), where p(x) is a polynomial. For each of the graphs, find the number of zeroes of p(x).

Theorem 6.2 :

Example 1:

If a line intersects sides AB and AC of a △ ABC at D and E respectively and is parallel to BC, prove that AD/AB = AE/AC (see Fig. 6.13.).

Example 2:

ABCD is a trapezium with AB || DC. E and F are points on non-parallel sides AD and BC respectively such that EF is parallel to AB (see Fig. 6.14). Show that AE/ED = BF/FC.

Example 3:

In Fig. 6.16, PS/SQ = PT/TR and ∠ PST = ∠ PRQ. Prove that PQR is an isosceles triangle.

Exercise 6.2

1. In Fig. 6.17, (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).

2. E and F are points on the sides PQ and PR respectively of a  PQR. For each of the following cases, state whether EF || QR :

(i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm

(ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm

(iii) PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm

3. In Fig. 6.18, if LM || CB and LN || CD, prove that AM/AB = AN/AD

4. In Fig. 6.19, DE || AC and DF || AE. Prove that BF/FE = BE/EC

5. In Fig. 6.20, DE || OQ and DF || OR. Show that EF || QR.

6. In Fig. 6.21, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.

7. Using Theorem 6.1, prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).

8. Using Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).

9. ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that AO/BO = CO/DO.

10. The diagonals of a quadrilateral ABCD intersect each other at the point O such that AO/BO = CO/DO. Show that ABCD is a trapezium.

6.4 Criteria for Similarity of Triangles

Theorem 6.3 :

Theorem 6.4 :

Theorem 6.5 :

Example 4:

In Fig. 6.29, if PQ || RS, prove that △ POQ ~ △ SOR.

Example 5:

Observe Fig. 6.30 and then find ∠ P.

Example 6:

In Fig. 6.31, OA . OB = OC . OD.

Show that ∠ A = ∠ C and ∠ B = ∠ D.

Example 7:

A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds.

Example 8:

Observe Fig. 6.30 and then find ∠ P.

In Fig. 6.33, CM and RN are respectively the medians of △ ABC and △ PQR. If △ ABC ~ △ PQR, prove that :

(i) △ AMC ~ △ PNR

(ii) CM/RN = AB/PQ

(iii) △ CMB ~ △ RNQ

Exercise 6.3

1. State which pairs of triangles in Fig. 6.34 are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form :

2. In Fig. 6.35, ∠ ODC ~ ∠ OBA, ∠ BOC = 125° and ∠ CDO = 70°. Find ∠ DOC, ∠ DCO and ∠ OAB.

3. Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using a similarity criterion for two triangles, show that OA/OC=OB/OD

4. In Fig. 6.36, QR/QS = QT/PR and ∠ 1 = ∠ 2. Show that △ PQS ~ △ TQR.

5. S and T are points on sides PR and QR of △ PQR such that ∠ P = ∠ RTS. Show that △; RPQ ~ △ RTS.

6. In Fig. 6.37, if △ ABE ≅ △ ACD, show that △ ADE ~ △ ABC.

7. In Fig. 6.38, altitudes AD and CE of  ABC intersect each other at the point P. Show that:

(i) △ AEP ~ △ CDP

(ii) △ ABD ~ △ CBE

(iii) △ AEP ~ △ ADB

(iv) △ PDC ~ △ BEC

8. E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that △ ABE ~ △ CFB.

9. In Fig. 6.39, ABC and AMP are two right triangles, right angled at B and M respectively. Prove that:

(i) △ ABC ~ △ AMP

(ii) CA/PA = BC/MP

10. CD and GH are respectively the bisectors of ∠ ACB and ∠ EGF such that D and H lie on sides AB and FE of △ ABC and △ EFG respectively. If △ ABC ~ △ FEG, show that: (i) CD/GH = AC/FG (ii) △ DCB ~ △ HGE (iii) △ DCA ~ △ HGF

11. In Fig. 6.40, E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD ⟂ BC and EF ⟂ AC, prove that △ ABD ~ △ ECF.

12. Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of  PQR (see Fig. 6.41). Show that △ ABC ~ △ PQR.

13. D is a point on the side BC of a triangle ABC such that ∠ ADC = ∠ BAC. Show that CA2 = CB.CD.

14. Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that △ ABC ~ △ PQR.

15. A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.

16. If AD and PM are medians of triangles ABC and PQR, respectively where △ ABC ~ △ PQR, prove that AB/PQ = AD/PM