Introduction and Basic Proportionality Theorem Video Lecture | Mathematics (Maths) Class 10

0:0:0: have you ever wondered how we were able
0:0:2: to measure the diameter of moon was it
0:0:5: measured using a measuring tape no it is
0:0:8: not possible to measure such a large
0:0:11: distance to estimate these values we
0:0:14: take indirect
0:0:16: measurements these measurements can be
0:0:18: anything such as congruence of figures
0:0:20: or similarity of figures in this video
0:0:24: we will try to understand what is
0:0:26: similarity and how can it be used in
0:0:28: daily life but before we start we should
0:0:31: recap some
0:0:33: Concepts a triangle is a figure which
0:0:35: has three sides and three angles the sum
0:0:39: of all angles of a triangle are
0:0:42: 180° we can say that two triangles are
0:0:45: congruent if they have equal sides and
0:0:48: angles let us see some congruent
0:0:51: figures circles with same radi square
0:0:55: with same side triangles with same sides
0:0:58: and angles
0:1:0: hence we can see that all congruent
0:1:3: figures are identical to each other but
0:1:6: let us see another example is there a
0:1:9: relation between these two
0:1:11: figures yes you might say that both of
0:1:14: them have the same shape but different
0:1:16: size this kind of relation is termed as
0:1:19: similarity of
0:1:21: figures so congruent figures have the
0:1:24: same shape and size similar figures have
0:1:27: same shape but different size
0:1:30: now mathematically we need to find a
0:1:32: relation for
0:1:34: similarity so let us analyze to similar
0:1:37: figures here we can see that their
0:1:40: angles are equal also their sides are
0:1:43: not equal but if we watch closely we
0:1:46: will observe that all the sides are in
0:1:48: the same ratio so to Define similarity
0:1:52: we must have to necessary conditions
0:1:55: equal
0:1:56: angles ratio of all sides must be equal
0:2:1: so a square and rectangle are not
0:2:3: similar because they have different
0:2:5: shape even though angles are equal the
0:2:8: ratio of their sides are different to
0:2:11: represent the similarity of figures we
0:2:14: use similar to symbol similarity of
0:2:17: triangles now that we have defined the
0:2:20: similarity of two figures we can try to
0:2:23: understand it in the case of
0:2:24: triangles two triangles are similar if
0:2:28: they have equal angles and ratio of
0:2:31: sides is
0:2:32: equal now let us try to derive a
0:2:35: relation of two
0:2:36: triangles it is called basic
0:2:38: proportionality theorem let us assume a
0:2:41: triangle a b c in which a line parallel
0:2:44: to side BC intersects other two sides a
0:2:48: and a c at d and e
0:2:50: respectively let us join b e and CD and
0:2:53: then draw DN perpendicular to AC and N
0:2:56: perpendicular to ab area of triangle is
0:3:0: equal to half into base into height so
0:3:3: putting these formula for different
0:3:6: triangles we get area of triangle a is
0:3:10: equal to half into a d into
0:3:15: n area of triangle bde is equal to half
0:3:19: into DB into
0:3:22: n area of triangle a is equal to half
0:3:26: into AE into
0:3:28: DM
0:3:31: area of triangle DEC is equal to half
0:3:34: into e c into
0:3:37: DM area of triangle a upon area of
0:3:41: triangle bde is equal to a upon
0:3:46: DB area of triangle a upon area of
0:3:50: triangle de C is equal to AE upon e
0:3:57: c now if we closely notice triangle bde
0:4:1: and triangle DEC have the same base and
0:4:3: against the same parallel hence the
0:4:6: value of perpendicular will be equal
0:4:10: hence they have equal area area of bde
0:4:13: is equal to area of DEC using previous
0:4:17: results and the fact that area of BD is
0:4:20: equal to area of DEC we conclude that a
0:4:24: d upon DB equal AE upon e c let us see
0:4:29: it in a triangle your ratio of a d upon
0:4:32: DB is equal to AE upon e
0:4:37: c if a line is drawn parallel to one
0:4:40: side of a triangle to intersect the
0:4:42: other two sides in distinct points the
0:4:45: other two sides are divided in the same
0:4:47: ratio this theorem is also called Tales
0:4:51: theorem this has great amount of uses in
0:4:55: mathematics hence we can utilize this
0:4:57: theorem for several cases in our real
0:5:1: life let us try to solve this simple
0:5:4: example here is a right triangle with d
0:5:7: e parallel to BC by basic
0:5:10: proportionality theorem a d upon DB
0:5:13: equal AE upon e c putting the values and
0:5:17: cross
0:5:19: multiplying the value of EC becomes 2
0:5:23: cm we can also derive its Converse
0:5:26: theorem which is if a line divides any
0:5:30: two sides of a triangle in the same
0:5:32: ratio then the line is parallel to the
0:5:34: third side this theorem can be proved by
0:5:37: taking a line de such that a d upon DB
0:5:40: equals AE upon e c and assuming that de
0:5:43: is not parallel to BC if de is not
0:5:47: parallel to b c draw a line D- parallel
0:5:50: to b c so a d upon DB equal ae- upon e-
0:5:56: C therefore AE upon e c equal = a e-
0:6:0: upon e- C but this is only possible when
0:6:4: both of these points
0:6:6: coincide hence if a d upon DB equals AE
0:6:9: upon e c is true then line de is
0:6:12: parallel to
0:6:14: BC let's revise the theorem BPT tells us
0:6:19: if a line is parallel to base then ratio
0:6:23: of sides cut is
0:6:25: equal however Converse of BPT tells us
0:6:28: that if a line Cuts two sides of a
0:6:30: triangle in equal ratio then it must be
0:6:33: parallel to the third
0:6:36: side remember how we discussed the
0:6:38: diameter of moon let us try to find that
0:6:43: now we take a round coin and place it
0:6:45: between our eyes and the moon let's say
0:6:48: the diameter of coin is 2
0:6:51: cm we have to make sure that it properly
0:6:54: covers the moon let us use lines to
0:6:57: represent the coin and the Moon
0:7:0: this line represents the coin with
0:7:1: diameter 2
0:7:3: cm this line represents Moon whose
0:7:6: diameter we need to find out let us call
0:7:9: it D upon setting the coin at an
0:7:12: appropriate distance we will be able to
0:7:14: cover all the Moon from it this distance
0:7:18: comes out to be 2.25 M from our research
0:7:22: we know that distance between Earth and
0:7:25: Moon is
0:7:26: 384,000
0:7:28: km now let us visualize a triangle for
0:7:33: ease we will cut the triangle into two
0:7:35: right
0:7:36: triangles we take one right triangle and
0:7:39: name it
0:7:40: ABC we name the points such that so that
0:7:43: de represents radius of coin and BC
0:7:46: represents the radius of the Moon the
0:7:49: coin will work as a line parallel to
0:7:52: base we can get the value of AE by using
0:7:55: Pythagoras Theorem a² equal e d s + a d
0:8:1: squ we get AE is equal to
0:8:5: 2.25 22222 1
0:8:8: M Now using Tales theorem we get a d
0:8:13: upon DB equal a upon EC we reciprocate
0:8:18: the terms so now we have DB upon a equal
0:8:23: EC upon AE we add one to both sides we
0:8:29: get
0:8:30: DB + a Upon A = E C + AE upon
0:8:35: AE which is nothing but AB upon a equal
0:8:39: AC upon AE we just need to put the
0:8:45: values we have values of ab a d and AE
0:8:52: 384400 upon 2.25 = AC upon 2.25 0022 2
0:9:0: 2221 on
0:9:2: calculations the value of AC becomes 38
0:9:6: 44 lakh
0:9:8: 3,796
0:9:9: M now to calculate the radius we use
0:9:12: Pythagoras Theorem AC c² = a² + b
0:9:18: c² b c² = a² -
0:9:23: a² putting the values we get BC equal
0:9:28: 1,78 8.44
0:9:30: km we get radius of moon as
0:9:35: 1,784 4
0:9:37: km to get diameter we double the radius
0:9:41: so diameter of moon equals
0:9:45: 34688 7
0:9:47: km hence Tales theorem has a wide
0:9:50: variety of
0:9:52: applications
0:9:54: conclusion we were able to derive and
0:9:56: explain the application of basic
0:9:58: proportion ality
0:10:1: theorem we hope that you have understood
0:10:3: the concept of basic proportionality
0:10:6: theorem now here's a secret on how you
0:10:9: can study effectively with the urv app
0:10:13: you can learn with chapter notes watch
0:10:15: video lectures and solve ncer based McQ
0:10:19: tests of this chapter on
0:10:21: urv and that's not all you also get
0:10:24: amazing courses for class 10 and much
0:10:26: more for your class 10 preparation thank
0:10:30: you