At our school we have extended math and regular. To go to extended, you have to pass your secondary 2 math over 65%. What we've learned in secondary 3math is the following but I'll only talk about one thing:
Theorems
for Segments within Triangles
In this lesson we'll learn properties of altitudes, medians,midsegments,
angle bisectors, and perpendicular bisectors of triangles. Allfour of these
types of lines or line segments within triangles are concurrent,meaning
that the three medians of a triangle share intersecting points, as dothe
three altitudes, midsegments, angle bisectors, and perpendicular bisectors.
The intersecting point is called the point of concurrency. The various
points of concurrency for these four types of lines or line segments
all
have special properties.
2.1 Altitudes of a Triangle
The lines containing the altitudes of a triangle meet at one point called
the orthocenter of the triangle. Because the orthocenter lies on the lines
containing all three altitudes of a triangle, the segments joining the
orthocenter
to each side are perpendicular to the side. Keep in mind that
the
altitudes themselves aren't necessarily concurrent; the lines that contain
the altitudes are the only guarantee. This means that the orthocenter isn't
necessarily in the interior of the triangle. There are two other common
theorems concerning altitudes of a triangle. Both concern the concept of
similarity. The first states that the lengths of the altitudes of similar
triangles follow the same proportions as the corresponding sides of the
similar triangles. The second states that the altitude of a right triangle
drawn from the right angle to the hypotenuse divides the triangle into
two similar triangles. These two triangles are also similar to the original
triangle. The figure below illustrates this concept.
2.2 Medians of a Triangle
Every triangle has three medians, just like it has three altitudes, angle
bisectors,
and perpendicular bisectors. The medians of a triangle are the
segments
drawn from the vertices to the midpoints of the opposite sides.
The
point of intersection of all three medians is called the centroid of the
triangle.
The centroid of a triangle is twice as far from a given vertex than it
is from the midpoint to which the median from that vertex goes. For example,
if a median is drawn from vertex A to midpoint M through
centroid
C, the length of AC is twice the length of CM. The centroid is 2/3
of
the way from a given vertex to the opposite midpoint. The centroid is
always
on the interior of the triangle.
Two more interesting things are true of medians. 1) The lengths of the
medians of similar triangles are of the same proportion as the lengths
of
corresponding
sides. 2) The median of a right triangle from the right angle to the hypotenuse
is half the length of the hypotenuse.
2.3Midsegments of a Triangle
The midsegment of a triangle is a segment whose endpoints are both
midpoints
of sides. Every triangle has three midsegments. The midsegment
of
a triangle is always parallel to the third side (the side whose midpoint
it doesn't include), and half as long as the third side.
2.4 Angle Bisectors of Triangles
The angle bisectors of a triangle intersect each other at a point called
the
incircle of the triangle. The incircle of a triangle is the same as the
center of a circle inscribed in a triangle. Every triangle can have exactly
one inscribed circle, whose center is the incircle of the triangle, which
is the point at which the angle bisectors of the triangle intersect. The
incircle, then, is equidistant from the three sides of the triangle--a
property that results from the inherent congruency of the radii of a circle.
Another property of angle bisectors has to do with the side opposite the
bisected angle. An angle bisector divides the side opposite the bisected
angle
into two segments that are of the same proportion as the other two
sides.
For example, in triangle ABC above, let the angle at vertex A be
bisected,
and let the bisector intersect BC at point D. BD/DC = BA/CA.
2.5 Perpendicular Bisectors of Triangles
The three perpendicular bisectors of a triangle intersect at one point
called the circumcenter of a triangle. The circumcenter is the center of
the circle circumscribed about the triangle and is equidistant from all
the
vertices
of the triangle. In this case the perpendicular bisectors of the sides
of
the triangles are lines, not segments. Therefore, the circumcenter of a
triangle
does not necessarily exist in the interior of the triangle. Often theperpendicular
bisectors of a triangle intersect outside the triangle.