Properties of TriangleHard

Question

Internal bisector of ∠A of triangle ABC meets side BC at D. A line drawn through D perpendicular to AD intersects the side AC at E and the side AB at F. If a, b, c represent sides of ᐃABC then

Options

A.AE is HM of b and c
B.AD =
C.EF =
D.the triangle AEF is isosceles

Solution

       
We have ᐃABC = ᐃABD + ᐃACD
⇒   bc sin A = cAD sin b × AD sin
⇒   AD =
Again AE = AD sec
= ⇒ AE is HM of b and c.
EF = ED + DF = 2DE = 2 × AD tan
=
As AD ⏊ EF and DE = DF and AD is bisector ⇒ AEF is isosceles.
Hence A, B, C and D are correct answers.

Create a free account to view solution

View Solution Free
Topic: Properties of Triangle·Practice all Properties of Triangle questions

More Properties of Triangle Questions

The centre of square ABCD is at origin. The vertex A is Z, then centroid of triangle BCD is at (where Z ∈ C) :...In a triangle ABC, if b2 + c2 = 3a2, then cot B + cot C − cot A is equals to -...bc cos2 + ca cos2 + ab cos2 =...In an equilateral triangle the inradius and the circumradius are connsected by -...The points P, Q and R are taken on the ellipse = 1 with eccentric angles θ, θ + α, θ + 2α then ...