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Name the special segment Options: Median, Angle Bisector,
Warm Up: Name the special segment Options: Median, Angle Bisector, Perpendicular Bisector, or Altitude
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Points of Concurrency
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Day 2: Example: CPR has vertices C(15, 1), P(9, 11), and R(2, 1). Determine the coordinates of point A on CP so that RA is the median of CPR. P R C
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In PQR, QS and RT are medians. If PT=3x-1, PS= 4x - 2, and
SR = 2x + 4, find TQ. What do we know about PS and SR?_______________________________ How does that help us?__________________________________ Once we find x, how can we find the measure of TQ?_________________________________
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Example: AM and CN are medians of ABC, and LX is a median of LMC. Find XM if BC=18. How does MC relate to BC?_______________________________ How does XM relate to CM?_______________________________ How does that help us?__________________________________
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In MOP, ∠O ≅ ∠MPO, m∠M = 40 degrees, and PN is an altitude.
Example: In MOP, ∠O ≅ ∠MPO, m∠M = 40 degrees, and PN is an altitude. Find m∠NPO. What are the measures of <MPO and <O? ______________________________ What is the measure of <PNO?________________ <NPO=_____________
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AC is an altitude of ABD, find m∠1.
Example: AC is an altitude of ABD, find m∠1. What is the measure of <ACB?____________________ How does that help us find <1?________________________ <1=________
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In LMG, MK is an angle bisector, m∠1=2n+10, m∠2=4n−32 and m∠L=60.
Example: In LMG, MK is an angle bisector, m∠1=2n+10, m∠2=4n−32 and m∠L=60. Find m∠G. What do we know about <1 and <2?_______________________ How does that help us find <G?______________________ <M = ______ <L = ______ <M + <L + <G = _______ <G = _____
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Warm Up: Name the point of concurrency where three medians intersect
Which two segments can intersect outside of a triangle? What is an incenter?
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Perpendicular Bisector Theorem:
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PQ = 7 PQ = RQ Perpendicular Bisector Theorem
Example: Find PQ. PQ = RQ Perpendicular Bisector Theorem 3x + 1 = 5x – 3 Substitution 1 = 2x – 3 Subtract 3x from each side. 4 = 2x Add 3 to each side. 2 = x Divide each side by 2. So, PQ = 3(2) + 1 = 7. PQ = 7
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Let’s remember what we know about Circumcenters…
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EX: Given that O is the Circumcenter of Triangle ABC,
what do we know to be true? OA= ? OA= OC OA= OB D E AF= ? AF= FB AD= ? AD= DC F CE= ? CE= EB
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EX: Given that O is the Circumcenter of Triangle ABC, find
the measure of AO. 16 9 AO= 16
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EX: CD is the perpendicular bisector of ABC.
Find the perimeter of triangle ABC. 8x-8 3x+2 3 PERIMETER: AC+BC+AB AB = 3+3 AB = 6 8+8+6 = 22 We know AC = BC AC = 8x-8 AC = 8(2)-8 AC = 16-8 AC=8 and BC = 8 So…8x-8 = 3x+2 5x-8 = 2 5x = 10 x = 2
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Angle Bisector Theorem:
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QS = SR Angle Bisector Theorem 4x – 1 = 3x + 2 Substitution
Example: Find QS. QS = SR Angle Bisector Theorem 4x – 1 = 3x + 2 Substitution x – 1 = 2 Subtract 3x from each side. x = 3 Add 1 to each side. Answer: So, QS = 4(3) – 1 or 11.
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Let’s remember what we know about Incenters…
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D is the incenter of Triangle ACF. Find the length of BD.
BD = DG BD = 5
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Let’s remember what we know about Medians and Centroids…
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In ΔXYZ, P is the centroid and YV = 12. Find YP and PV.
Centroid Theorem YV = 12 Simplify. YP + PV = YV Segment Addition 8 + PV = 12 YP = 8 PV = 4 Subtract 8 from each side. Answer: YP = 8; PV = 4
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WP = (1/3) WZ 18 = (1/3) WZ 18 *3 = WZ 54 = WZ WZ = 54 PZ = 36
In ΔXYZ, P is the centroid and WP= 18. Find PZ and WZ. WP = (1/3) WZ 18 = (1/3) WZ 18 *3 = WZ 54 = WZ WZ = 54 PZ = 36 PZ = (2/3) WZ PZ = (2/3) * 54 PZ = 36
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