# Triangles

**Exercise 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:

**Exercise 2**

**4.** In figure (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).

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

PE = 3.9 cm, EQ = 4 cm, PF = 3.6 cm and FR = 2.4 cm

PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = **9 cm**

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

**6.** In figure, if LM || CB and LN || CD, prove that

**7.** In figure, DE || AC and DF || AE. Prove that

**8.** In figure, DE || OQ and DF || OR. Show that EF || QR.

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

**10.**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).

**11.** 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).

**12.** ABCD is a trapezium in which AB II DC and its diagonals intersect each other at the point 0. Show that

**13.** The diagonals of a quadrilateral ABCD intersect each other at the point 0 such that Such that ABCD is a trapezium.

**Exercise 3**

**14.** State which pairs of triangles in figure, 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:

**15.**In figure, ΔODC – ΔOBA, ∠ BOC = 125° and ∠ CDO = 70°. Find ∠ DOC, ∠ DCO and ∠ OAB.

**16.** Diagonals AC and BD of a trapezium ABCD with AB || CD intersect each other at the point 0. Using a similarity criterion for two triangles, show that

**17.** In figure, and ∠1 = ∠ 2. Show that Δ PQS – Δ TQR.

**18.** S and T are points on sides PR and QR of a Δ PQR such that ∠ P = ∠ RTS. Show that Δ RPQ – Δ RTS.

**19.**In figure, if Δ ABE ≅ Δ ACD, show that Δ ADE — Δ ABC.

**20.** In figure, altitude AD and CE of a Δ ABC intersect each other at the point P. Show that:

(i) Δ AEP – Δ CDP

(ii) Δ ABD – Δ CBE

(iii) Δ AEP – Δ ADB

(iv) Δ PDC – Δ BEC

**21.** E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that Δ ABE — Δ CFB.

**22.** In figure, ABC and AMP are two right triangles, right angles at B and M respectively. Prove that:

**23.** CD and GH are respectively the bisectors of ∠ ACB and ∠ EGF such that D and H lie on sides AB and FE at Δ ABC and Δ EFG respectively. If Δ ABC — Δ FEG, show that:

**24.** In figure, E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD ⊥ BC and EF ⊥ AC, prove that Δ ABD – Δ ECF.

**25.**Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of a Δ PQR (see figure). Show that Δ ABC — Δ PQR.

**26.** D is a point on the side BC of a triangle ABC such that ∠ ADC = ∠ BAC. Show that CA^{2} = CB.CD.

**27.** 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 Δ ABC — Δ PQR.

**28. ** 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.

**29.** If AD and PM are medians of triangles ABC and PQR respectively, where Δ ABC – Δ PQR, prove that

**Exercise 4**

**30. ** Let Δ ABC – Δ DEF and their areas be, respectively, 64 cm^{2} and 121 cm^{2}. If EF = 15.4 cm, find BC.

**31.** Diagonals of a trapezium ABCD with AB || DC intersect each other at the point 0. If AB = 2CD, find the ratio of the areas of triangles AOB and COD.

**32.** In figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC at 0, show that

**33.** If the areas of two similar triangles are equal, prove that they are congruent.

**34.** D, E and F are respectively the mid-points of sides AB, BC and CA of A ABC. Find the ratio of the areas of A DEF and A ABC.

**35.** Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.

**36.** Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of the diagonals.

**Tick the correct answer and justify:**

**37.** ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Ratio of the areas of triangles ABC and BDE is:

(A) 2 : 1 (B) 1 : 2 (C) 4 : 1 (D) 1 : 4

**38.** Sides of two similar triangles are in the ratio 4 : 9. Areas of these triangles are in the ratio:

(A) 2 : 3 (B) 4 : 9 (C) 81 : 16 (D) 16 : 81

**Exercise 5**

**39.** Sides of triangles are given below. Determine which of them are right triangles. In case of a right triangle, write the length of its hypotenuse.

(i) 7 cm, 24 cm, 25 cm (ii) 3 cm, 8 cm, 6 cm

(iii) 50 cm, 80 cm, 100 cm (iv) 13 cm, 12 cm, 5 cm

**40.**PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR. Show that PM

^{2}= QM.MR.

**41.** In figure, ABD is a triangle right angled at A and AC ⊥ BD. Show that:

(i) AB^{2} = BC.BD

(ii) AC^{2} = BC.DC

(iii) AD^{2} = BD.CD

**42.** ABC is an isosceles triangle right angled at C. Prove that AB^{2} = 2AC^{2}.

**43.** ABC is an isosceles triangle with AC = BC. If AB^{2} = 2AC^{2}, prove that ABC is a right triangle.

**44.** ABC is an equilateral triangle of side 2a. Find each of its altitudes.

**45.** Prove that the sum of the squares of the sides of a rhombus is equal to the sum of squares of its diagonals.

**46.** In figure, 0 is a point in the interior of a triangle ABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. Show that:

(i) OA^{2} + OB^{2} + OC^{2} – OD^{2} – OE^{2} – OF^{2} = AF^{2} + BD^{2} + CE^{2}

(ii) AF^{2} + BD^{2} + CE^{2} = AE^{2} + CD^{2} + B F^{2}

**47.** A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from base of the wall.

**48.** A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attached to the other hand. How far from the base of the pole should the stake be driven so that the wire will be taut?

**49. **An aeroplane leaves an airport and flies due north at a speed of 1000 km pwe hour. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1200 km per hour. How far apart will be the two planes after hours?

**50. ** Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between the feet of the poles is 12 m, find the distance between their tops.

**51.** D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C. Prove that AE^{2} + BD^{2} = AB^{2} + DE^{2}.

**52.** The perpendicular from A on side BC of a A ABC intersects BC at D such that DB = 3CD (see figure). Prove that 2AB^{2} = 2AC^{2} + BC^{2}.

**53.** In an equilateral triangle ABC, D is a point on side BC such that BD = BC. Prove that 9AD^{2} = 7AB^{2}.

**54.** In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.

**55.** Tick the correct answer and justify: In A ABC, AB = 6Vii, cm, AC = 12 cm and BC = 6 cm. the angles A and B are respectively:

(A) 90^{0} and 30^{0} (B) 90^{0} and 60^{0} (C) 30^{0} and 90^{0} (D) 60^{0} and 90^{0}

**Exercise 6**

**56.** In figure, PS is the bisector of ∠ QPR of Δ PQR. Prove that

**57.** In figure, D is a point on hypotenuse AC of Δ ABC, BD ⊥ AC, DM ⊥ BC and DN ⊥ AB. Prove that:

(i) DM^{2} = DN.MC

(ii) DN^{2} = DM.AN

**58.**In figure, ABC is a triangle in which ∠ ABC > 90° and AD ⊥ CB produced. Prove that:

AC^{2} = AB^{2} + BC^{2} + 2BC.BD

**59.** In figure, ABC is a triangle in which ∠ ABC < 90° and AD ⊥ BC Prove that:

AC^{2} = AB^{2} + BC^{2} – 2BC.BD

**60.** In figure, AD is a median of a triangle ABC and AM ⊥ BC. Prove that:

**61.** Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.

**62.** In figure, two chords AB and CD intersect each other at the point P. Prove that:

(i) Δ APC — Δ DPB

(ii) AP.PB = CP.DP

**63.** In figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that:

(i) Δ PAC — Δ PDB

(ii) PA.PB = PC.PD

**64.** In figure, D is appoint on side BC of Δ ABC such that . Prove that AD is the bisector of ∠BAC,

**65.** Nazima is fly fishing in a stream. The tip of her fishing rod is 1.8 m above the surface of the water and the fly at the end of the string rests on the water 3.6 m away and 2.4 m from a point directly under the tip of the rod. Assuming that her string (from the tip of her rod to the fly) is taur, how much string does she have out (see figure)? If she pulls in the string at the rate of 5 cm per second, what will be the horizontal distance of the fly from her after 12 seconds?