6. Triangles

  • This page has links to complete syllabus of NCERT Chapter 6 Triangles based on syllabus recommended by CBSE based on class 10 maths.
  • There are 11 video lectures having explanations from basic to advance concepts, each and every NCERT solutions, questions and all important examples. You can access them directly from the lecture menu or just scroll down the page.
  • All lecture were uploaded on YouTube so for better learning experience here you will find link to YouTube Playlist of Chapter 6 Triangles.
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  • You can ask doubts in Triangles at the end of this page.

YouTube Playlist
6(A) || Exercise 6.1

00:00:20 Difference between congruent and similar figures

00:06:37 NCERT Exercise 6.1 Question 1

00:12:27 NCERT Exercise 6.1 Question 2

00:13:27 NCERT Exercise 6.1 Question 3

6(B) || Basic Proportionality Theorem

00:00:24 Proof of Basics Proportionality Theorem (Thales Theorem) Theorem 6.1 : If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

00:21:28 Proof of Converse of Basic Proportionality Theorem Theorem 6.2 : If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

00:30:48 NCERT Exercise 6.2 Question 1 1. In Fig. 6.17, (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).

6(C) || Exercise 6.2

00:00:48 NCERT Exercise 6.2 Question 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

00:08:48 NCERT Exercise 6.2 Question 3 In Fig. 6.18, if LM || CB and LN || CD, prove that AM/AB=AN/AD

00:18:28 NCERT Exercise 6.2 Question 4 In Fig. 6.19, DE || AC and DF || AE. Prove that BF/BE=FE/EC

00:23:18 NCERT Exercise 6.2 Question 5 In Fig. 6.20, DE || OQ and DF || OR. Show that EF || QR.

00:28:38 NCERT Exercise 6.2 Question 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.

00:32:30 NCERT Exercise 6.2 Question 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).

00:36:48 NCERT Exercise 6.2 Question 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).

6(D) || Exercise 6.2 (Q9, Q10) Theorem 6.3

00:01:02 NCERT Exercise 6.2 Question 9 ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show That AO/CO=BO/DO

00:10:54 NCERT Exercise 6.2 Question 10 The diagonals of a quadrilateral ABCD intersect each other at the point O such that AO/CO=BO/DO. Show that ABCD is a trapezium.

00:19:04 Detailed explanation of similar figures v/s congruent figures

00:27:24 Proof of AAA Similarity Criterion Theorem 6.3 : If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar.

6(E) || Theorem 6.4 Theorem 6.5

00:00:54 Proof of SSS Similarity Criterion Theorem 6.4 : If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similiar.

00:19:54 Proof of SAS Similarity Criterion Theorem 6.5 : If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.

6(F) || Exercise 6.3 (Q1 to Q9)

00:01:20 NCERT Exercise 6.3 Question 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 :

00:15:20 NCERT Exercise 6.3 Question 2 In Fig. 6.35, ∆ ODC ~ ∆ OBA, ∠ BOC = 125° and ∠ CDO = 70°. Find ∠ DOC, ∠ DCO and ∠ OAB.

00:22:10 NCERT Exercise 6.3 Question 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/OB=OC/OD

00:29:20 NCERT Exercise 6.3 Question 4 In Fig. 6.36, QR/QT=QS/PR and ∠ 1 = ∠ 2. Show that ∆ PQS ~ ∆ TQR.

00:34:50 NCERT Exercise 6.3 Question 5 S and T are points on sides PR and QR of ∆ PQR such that ∠ P = ∠ RTS. Show that ∆ RPQ ~ ∆ RTS.

00:38:10 NCERT Exercise 6.3 Question 6 In Fig. 6.37, if ∆ ABE ≅ ∆ ACD, show that ∆ ADE ~ ∆ ABC.

00:45:00 NCERT Exercise 6.3 Question 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

00:53:50 NCERT Exercise 6.3 Question 8 E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that ∆ ABE ~ ∆ CFB.

00:57:50 NCERT Exercise 6.3 Question 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/BC=PA/MP

6(G) || Exercise 6.3 (Q10 to Q16)

00:00:36 NCERT Exercise 6.3 Question 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/AC=GH/FG (ii) ∆ DCB ~ ∆ HGE (iii) ∆ DCA ~ ∆ HGF

00:13:07 NCERT Exercise 6.3 Question 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.

00:16:27 NCERT Exercise 6.3 Question 13 D is a point on the side BC of a triangle ABC such that ∠ ADC = ∠ BAC. Show that CA^2 = CB.CD.

00:23:07 NCERT Exercise 6.3 Question 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.

00:32:17 NCERT Exercise 6.3 Question 16

00:43:17 NCERT Exercise 6.3 Question 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.

00:52:37 NCERT Exercise 6.3 Question 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.

6(H) || Exercise 6.4 Theorem 6.6

00:00:30 Proof of Theorem 6.6 : The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

00:13:30 NCERT Exercise 6.4 Question 1 Let ∆ ABC ~ ∆ DEF and their areas be, respectively, 64 cm^2 and 121 cm2 . If EF = 15.4 cm, find BC.

00:17:00 NCERT Exercise 6.4 Question 2 Diagonals of a trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2 CD, find the ratio of the areas of triangles AOB and COD.

00:22:50 NCERT Exercise 6.4 Question 3 In Fig. 6.44, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that ar (ABC)/ar (DBC)=AO/DO

00:29:10 NCERT Exercise 6.4 Question 4 If the areas of two similar triangles are equal, prove that they are congruent.

00:34:50 NCERT Exercise 6.4 Question 5 D, E and F are respectively the mid-points of sides AB, BC and CA of ∆ ABC. Find the ratio of the areas of ∆ DEF and ∆ ABC.

00:44:30 NCERT Exercise 6.4 Question 6 Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.

00:52:50 NCERT Exercise 6.4 Question 7 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 its diagonals.

01:01:10 NCERT Exercise 6.4 Question 8 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

01:04:20 NCERT Exercise 6.4 Question 9 Sides of two similar triangles are in the ratio 4 : 9. Areas of these triangles are in the ratio

6(I) || Theorem 6.7 Theorem 6.8 Theorem 6.9

00:01:10 Proof of Theorem 6.7 : If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

00:24:23 Proof of Pythagoras Theorem Theorem 6.8 : In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

00:32:33 Proof of converse of Pythagoras Theorem Theorem 6.9 : In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

6(J) || Exercise 6.5 (Q1 to Q7)

00:01:06 NCERT Exercise 6.5 Question 1 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

00:07:46 NCERT Exercise 6.5 Question 2 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.

00:17:56 NCERT Exercise 6.5 Question 3 In Fig. 6.53, 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

00:26:36 NCERT Exercise 6.5 Question 4 ABC is an isosceles triangle right angled at C. Prove that AB^2=2AC^2

00:29:16 NCERT Exercise 6.5 Question 5 ABC is an isosceles triangle with AC = BC. If AB^2= 2AC^2, prove that ABC is a right triangle.

00:32:56 NCERT Exercise 6.5 Question 6 ABC is an equilateral triangle of side 2a. Find each of its altitudes.

00:38:06 NCERT Exercise 6.5 Question 7 Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.

6(K) || Exercise 6.5 (Q8 to Q17)

00:02:35 NCERT Exercise 6.5 Question 8 In Fig. 6.54, O 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+BF^2

00:14:35 NCERT Exercise 6.5 Question 9 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.

00:18:26 NCERT Exercise 6.5 Question 10 A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut?

00:24:45 NCERT Exercise 6.5 Question 11 An aeroplane leaves an airport and flies due north at a speed of 1000 km per 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 1 1/2 hours?

00:31:55 NCERT Exercise 6.5 Question 12 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.

00:36:05 NCERT Exercise 6.5 Question 13 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 .

00:41:35 NCERT Exercise 6.5 Question 14 The perpendicular from A on side BC of a ∆ ABC intersects BC at D such that DB = 3 CD (see Fig. 6.55). Prove that 2 AB^2 = 2 AC^2 + BC^2 .

00:47:05 NCERT Exercise 6.5 Question 15 In an equilateral triangle ABC, D is a point on side BC such that BD = 1/3 BC. Prove that 9AD^2=7AB^2

00:57:05 NCERT Exercise 6.5 Question 16 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.

01:01:25 NCERT Exercise 6.5 Question 17 Tick the correct answer and justify : In ∆ ABC, AB = 6 3 cm, AC = 12 cm and BC = 6 cm. The angle B is : (A) 120° (C) 90° (B) 60° (D) 45°

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