Basic Proportionality Theorem Explained | NCERT Maths Class 10

By Team Aakash Byju's

If a line is drawn parallel to one side of a triangle intersecting the other two sides at different points, then the other two sides are divided in the same ratio.

[{"selector":"#anim-04a86a5d-6d94-45b1-9fd7-d4d28f9317b3","keyframes":{"transform":["translate3d(-115.45741%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(0.4, 0.4, 0.0, 1)","fill":"both"}] [{"selector":"#anim-8452f86c-c003-4de7-b929-21ec34c1d9ed","keyframes":{"opacity":[0,1]},"delay":0,"duration":1300,"easing":"cubic-bezier(0.4, 0.4, 0.0, 1)","fill":"both"}] [{"selector":"#anim-bec6c54a-8ff7-4d08-ae25-e2d570c0940d","keyframes":{"transform":["scale(0.15)","scale(1)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(0.4, 0.4, 0.0, 1)","fill":"forwards"}]

Click Here To Attend Free Trail Class

[{"selector":"#anim-6cb62628-b8a5-4b10-a728-310b98487cbf","keyframes":[{"offset":0,"transform":"translate3d(0, -707.778%, 0)","easing":"cubic-bezier(.5, 0, 1, 1)"},{"offset":0.29,"transform":"translate3d(0, 0%, 0)","easing":"cubic-bezier(0, 0, .5, 1)"},{"offset":0.45,"transform":"translate3d(0, -199.0271736%, 0)","easing":"cubic-bezier(.5, 0, 1, 1)"},{"offset":0.61,"transform":"translate3d(0, 0%, 0)","easing":"cubic-bezier(0, 0, .5, 1)"},{"offset":0.71,"transform":"translate3d(0, -67.6635768%, 0)","easing":"cubic-bezier(.5, 0, 1, 1)"},{"offset":0.8,"transform":"translate3d(0, 0%, 0)","easing":"cubic-bezier(0, 0, .5, 1)"},{"offset":0.85,"transform":"translate3d(0, -25.409230200000003%, 0)","easing":"cubic-bezier(.5, 0, 1, 1)"},{"offset":0.92,"transform":"translate3d(0, 0%, 0)","easing":"cubic-bezier(0, 0, .5, 1)"},{"offset":0.96,"transform":"translate3d(0, -11.0413368%, 0)","easing":"cubic-bezier(.5, 0, 1, 1)"},{"offset":1,"transform":"translate3d(0, 0%, 0)","easing":"cubic-bezier(0, 0, .5, 1)"}],"delay":0,"duration":1300,"fill":"both"}] Download Aakash APP

Let us explore the theorem with the help of a diagram given below:

[{"selector":"#anim-eb946c4b-a4ea-467e-8a60-2da4794223aa","keyframes":{"transform":["translate3d(-115.45741%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(0.4, 0.4, 0.0, 1)","fill":"both"}] [{"selector":"#anim-ecf1a008-ded8-46f0-8541-f7a09ec555b5","keyframes":{"opacity":[0,1]},"delay":0,"duration":1300,"easing":"cubic-bezier(0.4, 0.4, 0.0, 1)","fill":"both"}] [{"selector":"#anim-2db2979d-d16e-4c35-8d11-11890c7ad092","keyframes":{"transform":["scale(0.15)","scale(1)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(0.4, 0.4, 0.0, 1)","fill":"forwards"}]

We notice that PQ is parallel to BC and intersects AB and AC at 2 distinct points.

[{"selector":"#anim-115bc83a-2983-4e10-8fbc-0aa656b49d59","keyframes":{"opacity":[0,1]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"both"}] [{"selector":"#anim-7ead6bf0-2343-49d7-b0bb-bf403bd15f65","keyframes":{"transform":["scale(0.3333333333333333)","scale(1)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"forwards"}]

The converse of the Basic Proportionality Theorem is also true.

[{"selector":"#anim-7ead3873-0490-45c9-bdbd-63390b99bce0","keyframes":{"opacity":[0,1]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"both"}] [{"selector":"#anim-a116c678-f833-4ae1-9fbb-7fedf319e22e","keyframes":{"transform":["scale(0.3333333333333333)","scale(1)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"forwards"}]

i.e., If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

[{"selector":"#anim-f5b5a8f7-5f47-4069-92da-ed1bb2b8b417","keyframes":{"opacity":[0,1]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"both"}] [{"selector":"#anim-681e21ae-ddde-4e1c-b55f-6cd0955e6415","keyframes":{"transform":["scale(0.3333333333333333)","scale(1)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"forwards"}]

The Thales Theorem

[{"selector":"#anim-196c634e-26a7-47be-b508-8cde1359d328","keyframes":{"opacity":[0,1]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"both"}] [{"selector":"#anim-ae72e187-26d0-4248-8024-91caa45b1655","keyframes":{"transform":["scale(0.3333333333333333)","scale(1)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"forwards"}] [{"selector":"#anim-7212fa3a-1b42-4c76-8032-e8a01b59438c","keyframes":{"opacity":[0,1]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"both"}] [{"selector":"#anim-fab6ac37-4953-4da5-9c9d-1624b4b6ba54","keyframes":{"transform":["scale(0.3333333333333333)","scale(1)"]},"delay":0,"duration":1300,"easing":"cubic-bezier(.3,0,.55,1)","fill":"forwards"}]