Chapter 5 Relationships within Triangles  Midsegments  Perpendicular bisectors - Circumcenter  Angle Bisectors – Incenter  Medians – Centroid  Altitudes – Orthocenter  Inequalities in one triangle  Inequalities in Two Triangles

Midsegment

Finding Lengths

Perpendicular Bisector Theorem  If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment

Converse of the Perpendicular Bisector Theorem  If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment

Using the Perpendicular Bisector Theorem  What is the length of QR?  How would you set up the problem?

Angle Bisector Theorem  If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle

Converse of the Angle Bisector Theorem  If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector.

Concurrency of Perpendicular Bisectors Theorem  The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices

Concurrency of Angle Bisectors Theorem  The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle

Concurrency of Medians Theorem

Altitude of a Triangle  The perpendicular segment from the vertex of the triangle to the line containing the opposite side  Can be on the inside, the outside, or a side of a triangle

Summary

Corollary to the Triangle Exterior Angle Theorem  The measure of an exterior angle is greater than the measure of each remote interior angles of a triangle

Applying the Corollary

Theorem  If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side

Theorem  If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle

Take Note  In order to form or construct a triangle the sum of the two shortest sides must be greater than the largest side.

Triangle Inequality Theorem

Find the Possible Lengths

The Hinge Theorem (SAS Inequality Theorem)  If two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle

Converse of the Hinge Theorem  If two sides of one triangle are congruent to two sides of another triangle, and the third sides are not congruent, then the larger included angle is opposite the longer third side.

Find the range of possible values for x

Chapter 7 Similarity  Ratios and Proportions  Similar Polygons  Proving Triangles Similar  Similarity in Right Triangles  Proportions in Triangles

Similar Figures  Have the same shape but not necessarily the same size  Is similar to is abbreviated by ~ symbol  Two Polygons are similar if corresponding angles are congruent and the corresponding sides are proportional

Finding Lenghts

Angle Angle Similarity (AA~)  If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar

Side Angle Side Similarity (SAS~)  If an angle of one triangle is congruent to an angle of a second triangle, and the sides that include the two angles are proportional then the triangles are similar

Side Side Side Similarity (SSS~)  If the corresponding sides of two triangles are proportional, then the triangles are similar

Are the Triangles Similar? If so write a similarity statement.

Geometric Mean  Proportions in which the means are equal  For numbers a and b, the geometric mean is the positive number x such that:  a = x x b  Then you cross multiply and solve for x

Theorem – Geometric Mean  The length of an altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse.

From the first example

What are the values of x and y?

Side-Splitter Theorem  If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally

Find the value of x

Corollary to the Side Splitter Thm  If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional

Triangle Angle Bisector Thm  If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle

Find the value of x

Chapter 8

Pythagorean Theorem  In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

45 – 45 – 90 Triangle  In a 45 – 45 – 90 Triangle, both legs are congruent and the length of the hypotenuse is √2 times the length of a leg.

30 – 60 – 90 Triangle  The length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is √3 times the length of the shorter leg.

Trigonometric Ratios

Find the value of w

Using Inverses  What is the measure of <X to the nearest degree?

Angle of Elevation and Angle of Depression  The angle of elevation and the angle of depression are congruent to each other.

Law of Sines  Relates the sine of each angle to the length of the opposite side  Use when you know AAS, ASA, or SSA  SSA is generally used for obtuse triangles

Law of Sines  Relates the sine of each angle to the length of the opposite side  Use when you know AAS, ASA, or SSA  SSA is generally used for obtuse triangles

Law of Cosines  Relates the cosine of each angle to the side lengths of the triangle  Use when you know SAS or SSS

 Find MN to the nearest tenth

Translating Figures  To translate a figure in the coordinate plane, translate each point the same units left/right and up/down.  For example each point of ABCD is translated 4 units right and 2 units down. So each (x, y) pair is mapped to (x+4, y-2)  Written as:

Properties of Reflections  Preserve Distance and Angle Measure  Reflections map each point of the preimage to one and only one corresponding point of its image

90 Degree Rotation

180 Degree Rotation

270 Degree Rotation

Dilations

Combinations

Find the Area of the Nonagon

 What is the area of a regular pentagon with 4in sides? Round your answer to the nearest square in.  A tabletop has the shape of a regular decagon with a radius of 9.5 in. What is the area of the tabletop to the nearest square inch?

Finding Area  Suppose you want to find the area of a triangle. What formula could you come up with to find the area of any triangle using a trig function  sinA = h/c  h = c sinA  A = ½(bc)sinA

What is the area of the triangle