Unit Essential Question: How do you use the properties of triangles to classify and draw conclusions?

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Presentation transcript:

Unit Essential Question: How do you use the properties of triangles to classify and draw conclusions?

Essential Question: What are the different classifications of triangles?

Classifying Triangles

Example:

Midsegment A midsegment joins the midpoints of two sides of a triangle.

Finding Lengths

Perpendicular Bisector Draw a line that is Perpendicular to this segment and bisects the segment: The perpendicular bisector of a segment is a line perpendicular to a segment at its midpoint.

Essential Question: What are the attributes of bisectors?

Perpendicular Bisector Theorem

DC Find the set of points equidistant from the White House and the Jefferson Memorial:

Distance from a point to a line: How far is it from this point to the line? How should we measure the distance?

Angle Bisector Theorem

Find FD.

Angle Bisector Theorem

Essential Question: Can you identify the median, altitude, angle bisector, and perpendicular bisector of a triangle?

Concurrency When three or more lines intersect at a common point, they are concurrent. The point of intersection of concurrent lines is the point of concurrency.

Concurrency in Triangles

Circumcenter & Incenter Point of concurrency of Perpendicular Bisectors forms the circumcenter, which is used to circumscribe a triangle. Point of concurrency of Angle Bisectors form the incenter, which is used to inscribe a triangle.

Median Will the medians be concurrent? Yes, at the centroid.

Altitude The altitude of a triangle is the perpendicular segment from a vertex to the line containing the opposite side. (Also called the height).

Are the Altitudes Concurrent? The point of concurrency is called the orthocenter.

Which special segments are pictured?

Practice

Essential Question: What is the relationship between the sides and opposite angles in a triangle?

What does this mean? Can we prove it?

Converse of Opposite Parts Theorem

Opposite Parts Theorem

List the Sides from Shortest to Longest

Triangle Inequality Theorem What is the special relationship?

Essential Question: How do you use the triangle inequality theorem to determine a range for the third side of a triangle?

Triangle Inequality Theorem

Can you make a triangle?

Range of Possible Lengths a – b < x < a + b