Q9 of 28 Page 112

The orthocentre of ΔABC is H. The mid-points of AH, BH and CH are respectively x, y and z. Prove that the orthocenter of ΔXYZ is also H.


Since , H is the orthocenter of the ΔABC


So, ADꞱBC , BEꞱAC and CFꞱAB


X, Y and Z are the midpoints of AH, BH and CH respectively


∴ XYZ is a triangle.



Hence, XZ||AC


So, ∠HOX=∠HEA=90°


Similarly, XY||AB


And ∠HPX=∠HFA=90°


And YZ||BC


So, ∠HQZ=∠HDC=90°


Hence, HO, HP and HQ are the altitudes of ∆XYZ


So, H is the orthocenter of ∆XYZ


More from this chapter

All 28 →
7

The medians AD, BE and CF in ΔABC pass through point G.

(a) If GF = 4 cm then find the value of GC.


(b) If AD = 7.5 cm then find the value of GD.


 8

ΔABC is an isosceles triangle in which AB = AC, the mid-point of BC is D. Prove that circumcenters, incentre, orthocenter and centroid all lie on line AD.

 10

How will you find the point in side BC of ΔABC which is equidistant from sides AB and AC.

 1

The point equidistant from the vertices of a triangle is called