chapter 5 relationships within triangles midsegments perpendicular bisectors - circumcenter angle...
TRANSCRIPT
Semester 2 Final Review
Chapters 5, 7 - 11
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 = xx 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?
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