Geometry Help?

Question

Show that if two altitudes of a triangle are perpendicular, the triangle is right-angled.

Help?

Answer 1

first let us consider a general triangle ABC

costruct altitudes from B & C to meet AC & AB at E & D respectively

let the altitudes intersect at point O

now in our case -> if altitudes are at 90

then

angle DOE = 90

also since they are altitudes

angle ADO = angle AEO = 90

in quadrilateral ADOE

sum of angles = 360

angle DAE + ADO + AEO + DOE = 360

angle DAE + 90 +90 +90 =360

angle DAE = 90

which implies the triangle is right angled

Answer 2

Let's mark the two vertices from which the two perpendicular heights emanate A and B. Let's denote height from A ha and height from B hb. Now the side a (opposite of vertex A) must be perpendicular to ha and pass through B. But according to axiom, there can be only one such a line, which is hb.
The same is for the other side of triangle (opposite of B)
Therefore hb = a and ha = b ==> a is perpendicular to b ==> the triangle is right-angled.

Answer 3

The heigth of a right-anglrd triangle is the length of one side of the 2 sides connected to the right-angle. So it is clear that the altitudes of a right-sides angle are perpendicular.

The opposite direction is more difficult
i leav that to the other posters.

I will give it a try.
let P1 be the altitude point, L1 its height vectpor and B1 its base.
then the other point of the triangle is on the line B1.

also
let P2 be the other altitude point, L2 its height vectpor and B2 its base.
then the other point of the triangle is on the line B2.

now given is that L1 and L2 are perpendicular thus L1 is B2 or L2 is B1 , so the points P1,P2 and the one on B1 ( or B2) forms a right angle. ( proof completed. )

Answer 4

An altitude means the height of a triangle. So, in order for two angles to meet at the highest point of the altitude then you must lengthen both until they intesect at a common point. When they meet at that highest altitude of the triangle, then, there you have it.. A right-angled triangle.

Hope yah understand it...

Answer 5

If you draw two altitudes, call H ithe point of cut.Call M and N the cuts with the sides they are pperpendicular to and A the vertix they form.
You have got a cuadrilateral HMAN.The sum of the angles is 360
degrees ,but the angle at M and N are right angles sum 180.
So if the angle at H is right (90) the angle at A also must be right