Solution :<\/h2>\n

Given<\/strong> : ABC is a triangle and D is point on BC such that \\(BD\\over CD\\) = \\(AB\\over AC\\)<\/p>\n

To Prove<\/strong> : AD is the bisector of \\(\\angle\\) BAC.<\/p>\n

Construction<\/strong> : Produce line BA to E such that line AE = AC. Join CE.<\/p>\n

Proof<\/strong> : In \\(\\triangle\\) AEC, since AE = AC, hence<\/p>\n

\\(\\angle\\) AEC = \\(\\angle\\) ACE\u00a0 \u00a0 \u00a0 (angles opposite to equal sides of triangle are equal)<\/p>\n

Now,\u00a0 \\(BD\\over CD\\) = \\(AB\\over AC\\)\u00a0 \u00a0 \u00a0(given)<\/p>\n

So,\u00a0 \\(BD\\over CD\\) = \\(AB\\over AE\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (AE = AC, by construction)<\/p>\n

\\(\\therefore\\)\u00a0 By converse of Basic Proportionality theorem(Thales Theorem),<\/p>\n

DA || CE<\/p>\n

Now, Since CA is a traversal, we have :<\/p>\n

\\(\\angle\\) BAD = \\(\\angle\\) AEC\u00a0 \u00a0 \u00a0 ……..(2)\u00a0 \u00a0 \u00a0 \u00a0 [corresponding angles]<\/p>\n

and\u00a0 \\(\\angle\\) DAC = \\(\\angle\\) ACE\u00a0 \u00a0 \u00a0 \u00a0……..(3)\u00a0 \u00a0 \u00a0 (alternate angles)<\/p>\n

Also,\u00a0 \\(\\angle\\) AEC = \\(\\angle\\) ACE\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(from 1)<\/p>\n

From (2) and (3),<\/p>\n

\\(\\angle\\) BAD = \\(\\angle\\) DAC<\/p>\n

Thus,\u00a0 AD bisects \\(\\angle\\) BAC.<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

Solution : Given : ABC is a triangle and D is point on BC such that \\(BD\\over CD\\) = \\(AB\\over AC\\) To Prove : AD is the bisector of \\(\\angle\\) BAC. Construction : Produce line BA to E such that line AE = AC. Join CE. Proof : In \\(\\triangle\\) AEC, since AE = AC, …<\/p>\n

In the figure, D is a point on side BC of triangle ABC such that \\(BD\\over CD\\) = \\(AB\\over AC\\). Prove that AD is the bisector of \\(\\angle\\) BAC.<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[43,912],"tags":[],"yoast_head":"\n