Exploring Relationships Within Triangles: Midsegments, Bisectors, and Inequalities
In this chapter, we explore the fundamental relationships within triangles through midsegments, perpendicular bisectors, and angle bisectors. Discover the Triangle Midsegment Theorem, which states that a segment connecting the midpoints of two sides is parallel and half the length of the third side. Learn about concurrent lines, including medians and altitudes, leading to key points like the centroid and orthocenter. We also delve into the Triangle Inequality Theorem and its implications for side lengths and angle measures, strengthening your understanding of triangle properties.
Exploring Relationships Within Triangles: Midsegments, Bisectors, and Inequalities
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Presentation Transcript
Chapter 5 Relationships Within Triangles
Warm Up: • Draw and cut out a triangle, any kind. • Fold one of the vertices such that it touches the opposite side and fold. • Measure the lengths a - f.
Midsegments • The segment connecting the midpoints of two sides Triangles have three midsegments
Triangle Midsegment Theorem • If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is half its length.
Example: Q and P are midpoints of two sides of the triangle. Find x and the perimeter of the triangle.
In ∆XYZ, M, N, and P are midpoints. The perimeter of ∆MNP is 60. Find NP and the perimeter of the triangle.
Triangles play a key role in relationships involving perpendicular bisectors and angle bisectors. B A
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.
C 5 A B 6 D Example One
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.
Example 2: Find FD C D B 5x 2x + 24 F FD = 2x + 24 = 2(8) +24 = 16 + 24 = 40 5x = 2x + 24 3x = 24 x = 8
1. Find the value of x. 2. Find CG. 3. Find the perimeter of quadrilateral ABCG. 6 8 48
Concurrent When three or more lines intersect in one point Point of Concurrency
Median The endpoints are a vertex of the triangle and the midpoint of the opposite side
Theorem 5-8 The medians of a triangle are concurrent at a point (the centroid) that is two thirds the distance from each vertex to the midpoint of the opposite side.
Find AB and AC if C is the centroid. A C 12 B
Checkpoint Use the Centroid of a Triangle The centroid of the triangle is shown. Find the lengths. 4. Find BE and ED, given BD=24. Find JG and KG, given JK=4. 5. PQ=10; PN=30 BE=16; ED=8 JG=12; KG=8 ANSWER ANSWER ANSWER Find PQ and PN, given QN=20. 6.
Altitude The perpendicular segment from a vertex to the line containing the opposite side (aka the height)
Altitude of a Triangle • Unlike angle bisectors and medians, an altitude of a triangle can be a side of a triangle, or it may lie outside the triangle. • The point of concurrency for the altitudes is called the orthocenter.
* Theorem 5-6 • The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. • The point of concurrency for the perpendicular bisectors is called the circumcenter.
* Theorem 5-7 • The bisectors of the angles of a triangle are concurrent at a point equidistant from its sides. • The point of concurrency for the angle bisectors is called the incenter.
Find the m<1. Complete the inequality statement: > >
Corollary to the Triangle Exterior Angle Theorem The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles.
Theorem 5-10: If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side.
Theorem 5-11: If two sides of a triangle are not congruent, then the longer side lies opposite the larger angle. Note: This is similar to 5-10, but reverses the order of the longer side and larger angle.
Checkpoint Order Angle Measures and Side Lengths Name the angles from largest to smallest. 1. ANSWER N;L;M 2. ANSWER Q;R;P 3. ANSWER U;S;T
Checkpoint Name the sides from longest to shortest. Order Angle Measures and Side Lengths 4. ANSWER GH;JG;JH 5. ANSWER DE;EF;DF AC;AB;BC 6. ANSWER
Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Checkpoint Use the Triangle Inequality Can the side lengths form a triangle? Explain. 5, 7, 13 7. ANSWER No; 5 + 7 < 13. 6, 9, 12 8. Yes; 6 + 9 > 12, 6 + 12 > 9,and 9 + 12 > 6. ANSWER 10, 15, 25 9. ANSWER No; 10 + 15 = 25.
The lengths of two sides of a triangle are given, what are the possible lengths for the third side?6”, 10” The difference of the two given sides is the minimum value that x can be. The sum of the two given sides is the maximum value. Hint: what are the three cases? The limits: 4 < x < 16
Coordinate Proofs: Given: R is the midpoint of S is the midpoint of Prove: Place the triangle in a convenient spot on the coordinate plane (typically with a vertex at (0, 0).
To find the coordinates of R and S, use the midpoint formula: Q (b, 0) and P (c, a) : O (0, 0) and P (c, a) :
To show , prove their slopes are equal. 3. Use the coordinates to prove the desired properties. = 0 = 0 Therefore, since the slopes are the same, the lines are parallel.
To show , prove using their distances. 3. Use the coordinates to prove the desired properties.
Warm Up: Write a coordinate proof showing opposite sides of a rectangle are parallel.
Objective One • The first objective in this section is to use inequalities involving angles of triangles.
