Constructions

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Constructions
Orthocenters
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Review of Prerquisites
To construct a perpendicular line through either
an external or an internal point the technique is
called
A Smile and a Fish.
Lets begin !
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Construct an Altitude from C.
The method is
A Smile
and
A Fish
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A Smile
Swing arc below from C
Are you Smiling ?
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Construct Fish
Swing arc from point x.
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Construct Fish
Swing arc from point y.
y
X
Connect the 2 points of intersections of the 2
arcs.
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After shortening the line.
Voila ! Altitude from C
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Rotate triangle clockwise to rest on side
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To find the altitude from A, we repeat the
process.
A Fish
A Smile
and
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Construct The Smile
Swing arc below from A
Are you Smiling ?
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Construct Fish
Swing arc from point x.
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Construct Fish
Swing arc from point y.
Connect the 2 points of intersections of the 2
arcs.
Shorten the line to the altitude distance.
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Voila . The Altitude
Rotate triangle clockwise to rest on side
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Start With a Smile From Point B
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Construct Fish
Swing arc from point x.
x
y
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Construct Fish
Swing arc from point y.
Connect the 2 points of intersections of the 2
arcs.
Shorten the line to the altitude distance.
x
y
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Voila . The Altitude
And the Orthocenter
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More Practice
Lets find the orthocenter of an obtuse triangle.
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Find Altitude from B
A Smile
Swing arc below from B
Note that we have only 1 intersection point with
the smile or arc.
We will need to extend segment AC.
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X
y
Now we have two intersection points.
Now we can start the Fish.
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X
y
Swing arc from point y.
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Now swing the arc from x.
X
y
Use the intersections of the arcs to draw a
line.
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Shorten line to the altitude distance.
Rotate triangle clockwise to rest on to side
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Find Altitude from A
Swing arc below from A
A Smile
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Here is the smile.
y
x
Now, we can make the Fish.
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x
y
Swing arc from point x.
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Swing arc from point y.
y
x
Use the intersections of the arcs to draw a
line.
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Shorten line to the altitude distance.
Note that the altitudes are not concurrent.
But their lines are concurrent.
The orthocenter is not always inside the triangle.
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It is only necessary to draw 2 altitudes.
The third altitude would only intersect the
other lines at the same point.
We will now look at several examples of
orthocenters to solidify your understanding.
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The orthocenter is inside the triangle for acute
angles.
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The orthocenter is at the vertex of the right
angle for right triangles.
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The orthocenter is outside the triangle for
obtuse angles.
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Summary
To construct a perpendicular line through either
an external or an internal point the technique is
called
A Smile and a Fish.
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Summary
1. An altitude is a perpendicular segment from
the vertex to the opposite line extended.
2. The altitude does not have to be in the
triangle. It could be ON or OUTSIDE the triangle.
3. The altitudes are not always concurrent. But
the lines containing the altitudes are always
concurrent.
4. All altitudes are made with a SMILE and a
FISH.
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Cest fini.
Good day and good luck.
Thats all folks.
A Senior Citizen Production