# Altitude of a Triangle: Showing indeed perpendicular

$$\textbf{Question:}$$ Why can we drop a perpendicular $$h$$ between the two points along the base of the triangle below where $$\alpha, \beta\leq 90^\circ$$?

The reason I'm wondering this is because often times when dealing with isosceles triangles, I see $$h$$ is often constructed by connecting $$M$$ to the top vertex by defining $$M$$ to be the midpoint which implies the angle along the base is indeed ninety degrees by using the perpendicular bisector theorem. So, do I need to define where $$M$$ is first for this example to show construction of $$h$$ will be indeed $$90^\circ$$ or no? That's where I am getting confused..

Here is a picture.

• So the question is 'why can we draw an altitude' or 'how do you draw an altitude precisely'? I don't really understand it... – Dr. Mathva Feb 9 '19 at 18:00
• Why can we draw an altitude. Why is it feasible? Is it due to betweenness? I'm not concerned about how to draw it with a compass and a straight edge. – W. G. Feb 9 '19 at 18:02
• I DON'T want the compass. – W. G. Feb 9 '19 at 18:05
• What do you think geometry is about? Why do you reject the compass and straightedge? – David K Feb 9 '19 at 18:07
• I'm not concerned about the construction of it by an example. I want the proof. – W. G. Feb 9 '19 at 18:11

Let we say about $$\Delta ABC$$ and let $$M$$ placed on the line $$AB$$ such that $$A$$ placed between $$M$$ and $$B$$.

Thus, $$\measuredangle CAM<90^{\circ},$$ which gives $$\alpha>90^{\circ},$$ which is a contradiction.

By the same way we can get a contradiction if $$B$$ placed between $$M$$ and $$A$$.

Id est, $$M$$ placed on the side $$AB$$.

• Rather, we always have $\measuredangle CAM<90^{\circ}$, but if $A$ were between $M$ and $B$, $\angle CAB$ would be the supplement of $\angle CAM$ rather than the same angle. – Misha Lavrov Feb 9 '19 at 18:19
• @Misha Lavrov If $M\in AB$ it's not always so! For example: $M\equiv A$. We have no an angle! – Michael Rozenberg Feb 9 '19 at 18:23
• So, I made a picture above. Why do we know $\measuredangle CAM<90^\circ$? – W. G. Feb 9 '19 at 18:35
• @W. G Because $\Delta CAM$ is a right-angled triangle. – Michael Rozenberg Feb 9 '19 at 18:36
• Ohhhhhhh!!!! I get it thank you! – W. G. Feb 9 '19 at 18:38