Advertisements
Advertisements
Question
Prove that sum of any two sides of a triangle is greater than twice the median with respect to the third side.
Advertisements
Solution
Given: In triangle ABC with median AD,
To proof: AB + AC > 2AD
AB + BC > 2AD
BC + AC > 2AD
Producing AD to E such that DE = AD and join EC.
Proof: In triangle ADB and triangle EDC,
AD = ED ...[By construction]
∠1 = ∠2 ...[Vertically opposite angles are equal]
DB = DC ...[Given]
So, by SAS criterion of congruence]
ΔADB ≅ ΔEDC
AB = EC ...[CPCT]
And ∠3 = ∠4 ...[CPCT]
Again, in triangle AEC,
AC + CE > AE ...[Sum of the lengths of any two sides of a triangle must be greater than the third side]
AC + CE > AD + DE
AC + CE > AD + AD ...[AD = DE]
AC + CE > 2AD
AC + AB > 2AD ...[Because AB = CE]
Hence proved.
Similarly, AB + BC > 2AD and BC + AC > 2AD.
APPEARS IN
RELATED QUESTIONS
ABC is a right angled triangle in which ∠A = 90° and AB = AC. Find ∠B and ∠C.
ΔABC and ΔDBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see the given figure). If AD is extended to intersect BC at P, show that
- ΔABD ≅ ΔACD
- ΔABP ≅ ΔACP
- AP bisects ∠A as well as ∠D.
- AP is the perpendicular bisector of BC.
AD is an altitude of an isosceles triangles ABC in which AB = AC. Show that
- AD bisects BC
- AD bisects ∠A
Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ΔPQR (see the given figure). Show that:
- ΔABM ≅ ΔPQN
- ΔABC ≅ ΔPQR
In two right triangles one side an acute angle of one are equal to the corresponding side and angle of the other. Prove that the triangles are congruent.
In the following figure, BA ⊥ AC, DE ⊥ DF such that BA = DE and BF = EC. Show that ∆ABC ≅ ∆DEF.
ABC is an isosceles triangle in which AC = BC. AD and BE are respectively two altitudes to sides BC and AC. Prove that AE = BD.
In a right triangle, prove that the line-segment joining the mid-point of the hypotenuse to the opposite vertex is half the hypotenuse.
Two lines l and m intersect at the point O and P is a point on a line n passing through the point O such that P is equidistant from l and m. Prove that n is the bisector of the angle formed by l and m.
ABC is a right triangle such that AB = AC and bisector of angle C intersects the side AB at D. Prove that AC + AD = BC.
