Properties of Triangles. Vocabulary Words 1.Equidistant 2.Locus 3.Concurrent 4.Point of concurrency...
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Transcript of Properties of Triangles. Vocabulary Words 1.Equidistant 2.Locus 3.Concurrent 4.Point of concurrency...
![Page 1: Properties of Triangles. Vocabulary Words 1.Equidistant 2.Locus 3.Concurrent 4.Point of concurrency 5.Circumcenter 6.Median 7.Centroid 8.Altitude 9.Orthocenter.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649e795503460f94b794a2/html5/thumbnails/1.jpg)
Properties of Triangles
![Page 2: Properties of Triangles. Vocabulary Words 1.Equidistant 2.Locus 3.Concurrent 4.Point of concurrency 5.Circumcenter 6.Median 7.Centroid 8.Altitude 9.Orthocenter.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649e795503460f94b794a2/html5/thumbnails/2.jpg)
Vocabulary Words
1. Equidistant2. Locus3. Concurrent4. Point of concurrency5. Circumcenter6. Median7. Centroid8. Altitude9. Orthocenter10. Incenter
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Perpendicular and Angle Bisectors
Equidistant – when a point is the same distance from two o r more objects.
Theorems: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.
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Example
Find each measure.
A. PB B. AB
C. AD
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Distance and Angle Bisectors
Locus – a set of points that satisfies a given condition exp: The perpendicular bisector of a segment can be defined as the locus of points in a plane that are equidistant from the endpoints of the segment.
Angle Bisector Theorem – If a point is on the bisector of an angle, then it 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 it is on the bisector of the angle.
![Page 6: Properties of Triangles. Vocabulary Words 1.Equidistant 2.Locus 3.Concurrent 4.Point of concurrency 5.Circumcenter 6.Median 7.Centroid 8.Altitude 9.Orthocenter.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649e795503460f94b794a2/html5/thumbnails/6.jpg)
Example
Find each measure
A. ED
B. given that = 112
C. ,
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Bisectors of Triangles
Concurrent – when three or more lines intersect at one point.
Point of concurrency – the point where three or more lines intersect.
Circumcenter of the triangle – the point of concurrency of the three perpendicular bisectors of a triangle.
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Cirmcumcenter Theorem
The circumcenter of a triangle is equidstant from the vertices of the triangle.
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Example
, , and are perpendicular bisectors of . Find HZ.
Your turn1. GM2. GK3. JZ
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Incenter Theorem
A triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the incenter of the triangle.
Incenter Theorem – The incenter of a triangle is equidistant from the sides to the triangle.
Unlike the circumcenter, the incenter is always inside the triangle.
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Example
JV and KV are angle bisectors of
A. The from V to KL.
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Your Turn
QX and RX are angle bisectors of Find each measure.
1. The distance from X to PQ 19.22. m 52
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Name Type Point of Concurrency
Perpendicular Bisector A line segment with the midpoint of a side as an end point
Circumcenter
Angle Bisector Bisects an angle on the interior of the triangle into two congruent angles
Incenter
Median A line segment with endpoints from a vertex and the midpoint of the opposite side
Centroid
Altitude Is a line segment from a vertex that is perpendicular to the side opposite the vertex
Orthocenter
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Inequalities and Triangles
Page 247
Resource Book page 17
Review Exterior Angle TheoremPage 248 Exterior Angle Inequality Theorem
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One Triangle Inequality
Page 261
Resource Book page 29
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Answer: Because the sum of two measures is not greater than the length of the third side, the sides cannot form a triangle.
Determine whether the measures and
can be lengths of the sides of a triangle.
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Determine whether the measures 6.8, 7.2, and 5.1 can be lengths of the sides of a triangle.
Check each inequality.
Answer: All of the inequalities are true, so 6.8, 7.2, and 5.1 can be the lengths of the sides of a triangle.
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In and Find the range of the third side.
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Inequalities Involving Two Triangles
Resource Book page 35