 Hello friends welcome again to another session on Gems of Geometry. In this session we are going to tackle another theorem which suggests or which says rather that tangent square right so square of a tangent in this case pt is the tangent at point t to the circle with the center o right so this is the tangent pt now square of pt is equal to pa into pa dash what is pa pa dash so a a dash is the chord and we have produced a a dash towards one side so this is the point p from where pt has been drawn as a tangent to the circle okay so if that is this is the diagram then pt squared will be equal to pa and pa dash so what i've done here is i have calculated i have first of all drawn you know the entire diagram with two scale actually so it's a geojibra software where i have drawn the circle i've drawn the tangent and the chord a a dash and produced it to point p and we have also joined pa and produced it to cut the circle at b dash and b so b and b dash so b and b dash is the surface e d is point this point yeah so b and b dash uh is it that b b dash is a diameter okay so pv b dash okay so i hope the construction is clear so pt is the tangent a a dash is the chord bb dash is the diameter all of them produced together on one side are meeting at point p okay and now uh let's verify whether this particular relationship actually holds so what i've done is i have calculated pt squared then i've also calculated pa into pa dash and i have also calculated pb into pb dash so let's verify whether this particular relationship is valid so i'm going to move the point a dash in such a way all the other points move relative to point a dash so that different different scenarios are built and see whether these values are same so please focus on these three values here right now it is 85.3 85.3 85.3 let's try and move the point a dash so i am moving point a dash and you can see wherever i stop the values all the three values are same is it so it looks like it is valid so far right so hence i'm moving point a through the diagram and you can see the values are not changing they are all same at any point okay at any point now i've shifted over to the other side see in any case if you see the values are the same are the same okay so this is what this theorem is talking you know uh telling us now i've also marked two angles here if you see alpha which is pta and pa dash t both are also equal why why is this because we have learned the theorem that the angle made between the tangent and the chord at the point of tangency that is angle pta is the angle made by the tangent pt and at chord will be equal to the angle subtended by the chord at on the other side of the circle right so at should not make an angle on this this side of you know the circle it should be on the other side in uh other that you know the other side means other to what other to the tangent so the angle made by the chord at must not be in the same side with the tangent it should be on the other side so if you see the angle made by the chord at on the other side of the tangent pt is a a dash t and hence it will always be same so can you can you see that so as i'm changing the position of a dash the value of alpha and beta remain the same okay so this is what we observed now let's see the proof so hence consider triangle pa t so pa and t and also consider pta dash so p pa dash okay so the moment uh we see such kind of a relationship where segments are being multiplied or we kind of get a hint that it must be related to similarity of triangles so the moment i choose these two triangles once again which two triangles pta and pa dash t okay now if you see in both these triangles angle p is common which p so t pa is the angle in triangle pa t and a dash p t is the angle in pa dash t triangle right so this this angle here is common to both the triangles now the other two angles are same anyway so pta is equal to pa dash t which we have already you know observed why is that because angle made by angle made between the chord and the tangent at the point of tangency is equal to angle subtended by the chord on the other side of the circle right so by these two angles if these two angles are equal then the given two triangles that is pa t and pa dash once again pa t this one this smaller one here and pa t a dash this triangle this bigger one here this pa dash are both similar and if they are similar then i can equate their corresponding side ratios okay so let's say you consider these so this is the thing pa t is is similar to pta dash so if i write pt on the left hand side then the corresponding side to pt will be pa dash so i've written that right so pt upon pa dash and on the other side if pt is this side here then the corresponding side to pt is pa so that means pt by pa dash is equal to pa by pt right by similar parts of similar triangle or similar sides or other sides of two similar triangles are proportional so by this logic if i cross multiply i'll get pt square is equal to pa times pa dash and that is what we intended to prove so this is what the proof is for this particular theorem so that once again any tangent pt squared will be nothing but equal to pa times pa dash where a a dash is a chord now if you notice bb dash is also a chord though it's a diameter so diameter is the largest chord so hence it will work with diameters as well so hence pt square will be also equal to pb into pb dash right so that's what we saw it here okay so i hope you understood this theorem