Vocabulary
Concurrent Lines – when three or more lines intersect in one point.
Point of concurrency– the point at which 3 or more lines intersect.
Vocabulary
Circumcenter of the triangle– the point of concurrency of the perpendicular bisectors.
Circumcenter
The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.
Theorem
Since the vertices of the triangle are equidistant from the circumcenter, we can draw a circle around the triangle or circumscribe the triangle.
The center of the circle is the circumcenter of the triangle.
Theorem
The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.
We can now inscribe a circle in the triangle since the incenter is equidistant from the sides.
The center of the circle is the incenter of the triangle.
Vocabulary
Altitude of a Triangle– a perpendicular segment from a vertex to the line containing the side opposite to the vertex. Orthocenter
Vocabulary
Median of a Triangle– a segment whose endpoints are a vertex and the midpoint of the opposite side.
Centroid
FYI
The centroid is the center of gravity of the triangle. If you cut out the triangle and paper-fold to determine the centroid, you can balance the triangle on the tip of your pencil point.
Theorem
The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side.
Cut out your triangles.
Yellow Paper - Median•Draw an acute triangle.•Cut it out.•Determine the midpoint of each side by folding corner to corner. Then fold the triangle from the midpoint of each side to the vertex opposite it.•Ask Mrs. Slifko to see the model in my notebook if you need to .•Label the point of concurrency the centroid. Try to balance your triangle on the tip of your pencil.•Put any other theorems on the triangle.
Cut out your triangles.
Green Paper- Altitude•Draw an acute triangle.•Cut it out.•Fold the triangle so that you create an altitude extending from all three vertices.•Ask Mrs. Slifko to see the model in my notebook if you need to .•Label the point of concurrency the orthocenter. •Write any necessary information on the triangle.
Example:
A(2,9)
B(-2,3) C(6,-1)
How can we find the slope of altitude AD?
D
We know that altitudes are at right angles to the
opposite sides…therefore, if we can find the slope of BC, we can use opposite
reciprocals to find the slope of AD.
Example:
A(2,9)
B(-2,3) C(6,-1)
How can we find the equation of altitude AD?
D
If you know a point on the altitude , namely
A(2,9),and the slope of AD, then you can use the point –slope formula to find the equation of altitude AD.
Example:
A(3,9)
B(-2,3) C(6,-1)
How can we find the slope of median AD?
D
Since we know that a median hits the midpoint,
we know that D is the midpoint of BC. Once you find the midpoint, you can use the slope formula to calculate the slope of AD.
Example:
A(3,9)
B(-2,3) C(6,-1)
How can we find the equation of median AD?
D
Since you know a point on the median line and a
slope, you can use point-slope formula to determine the equation of median AD.