CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Prove that the sum of three altitudes of a triangle is less than the perimeter of the triangle.

Open in App
Solution

Given: A triangle ABC in which ADBC,BEAC and CFAB.
To prove:
AD+BE+CF<AB+BC+CA
or AD+BE+CF< Perimeter of ΔABC
Proof: As we know that from all the segments that can be drawn to a given line, from a point not lying on it, the perpendicular line segment is the shortest one.
ADBC
AB>AD and AC>AD
AB+AC>2AD …(i)
BEAC
BC>BE and BA>BE
BC+BA>2AB …(ii)
Also CFAB
AC>CF and BC>CF
AC+BC>2CF …(iii)
Adding (i), (ii) and (iii), we get
(AB+AC)+(AB+BC)+(AC+BC)>2AD+2BE+2CF
2(AB+BC+CA)>2(AD+BE+CF)
AB+BC+CA>AD+BE+CF
Similarly, Perimeter of ΔABC>AD+BE+CF
Hence proved.
1870001_1878755_ans_01cc2d4f269946afa8de5e809e54a4cf.png

flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Criteria for Congruency
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon