Math Nation Section 7 Topics 3 8: Special Segments in a Triangle Notes (7.1 7.4 Extension) Proportionality caused by a Parallel Segment Ex 1) Ex 2) Ex 3) How do we know that ΔABG ~ ΔACF ~ ΔADE? P a g e 1
Math Nation Section 7 Topics 3 8: Special Segments in a Triangle Notes Midsegment a midsegment of a triangle is a segment that connects the midpoints of two sides of the triangle Special Properties: 1) Midsegment ll Base 2) Midsegment = ½ Base or 2(Midsegment) = Base Midsegment Triangle Formed by the 3 midsegments of the triangle Special Properties: 1) Perimeter of ΔDEF = ½ ΔABC 2) Area of ΔDEF = ¼ ΔABC If X and Y are Midpoints Then: 1) XY II MN 2) 2(XY) = MN 3) PXY PMN and PYX PNM (corresponding angle theorem) Algebraic Proof: If M and N are Midpoints, Prove that MN II RS and MN = ½ RS. Slope of MN: Slope of RS: Length of MN = Length of RS = P a g e 2
Perpendicular Bisector (Point of Concurrency ) the line that is perpendicular to the segment at its midpoint Perpendicular Bisector Theorem: if a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If CP AB Then Ex) Find the circumcenter of R (-4,-3) Q(4,5) S(4,-3). Locations of the Circumcenter based on Triangle Angle Classification P a g e 3
Angle Bisector (Point of Concurrency ) a ray that divides an angle into two congruent adjacent angles Angle Bisector Theorem: if a point lies on the bisector of an angle, then it is equidistant from the two sides of the angle. Distance from a point to a line must be perpendicular If AD bisects BAC Then D is equidistant from B and C. Error Analysis: Is QS an angle bisector? Real World Application of Circumcenter and Incenter When you want to find a point that is equidistant from 3 locations, find the When you want to find a point that is equidistant from 3 sides of a triangle, find the P a g e 4
Median (Point of Concurrency ) a segment from a vertex to the midpoint of the opposite side Centroid is the center of gravity for a triangle Centroid Properties: Median is divided in a 2:1 ratio Vertex to Centroid = 2 (Centroid to Midpoint) BP = 2(PF) AP = 2(PE) CP = 2(PD) Vertex to Centroid = 2/3 (Median) BP = 2/3 (BF) AP = 2/3 (AE) CP = 2/3 (CD) Centroid to Midpoint = 1/3 (Median) PF = 1/3 (BF) PE = 1/3 (AE) PD = 1/3 (CD) Altitude (aka the ) (Point of Concurrency ) an altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side (if it were extended) In the formula for the area of a triangle, A = ½bh, you can use the length of any side for the base b as long as you find the altitude or the height that is perpendicular to that base from the opposite vertex. Locations of the Orthocenter in Different Triangle Angle Classifications P a g e 5
Equilateral and Isosceles Triangle (special segments within) From any vertex Perpendicular Bisector Angle Bisector Median Altitude are the same segment. From vertex to base Perpendicular Bisector Angle Bisector Median Altitude are the same segment. SKILL BUILDING DRILLS: 1) Find HG 2) Find EG 3) Find SR 4) Write an equation of the perpendicular bisector of the segment with endpoints P( 1, -5) and Q(3, -1). Show all work. 5) Find x. 6) Find m GFH P a g e 6
IDENTIFYING DRILL: Given the following pictures and markings, identify each of the following as (a) an angle bisector, (b) a perpendicular bisector, (c) an altitude, or (d) a median. List all that apply. 1. 1. 2. 3. 2. 3. 4. 4. 5. 6. 5. 6. 7. 8. 7. 8. 9. 9. APPLICATION PROBLEMS: 1) Three snack carts sell frozen yogurt from points A, B, and C outside a city. Each of the three carts is the same distance from the frozen yogurt distributor. Explain how you would find the location for the distributor and draw a sketch on the triangle provided. Point of Concurrency needed: Explain: P a g e 7
2) Find the coordinates of the circumcenter of ΔABC with the given vertices: A(0, 3) B(0, 1) C(6, 1) AB: Midpoint Slope BC: Midpoint Slope AC: Midpoint Slope Circumcenter Coordinates: 3) Find the coordinates of the circumcenter of ΔRST with the given vertices: R( 2, 5) S( 6, 5) T( 2, 1) RS: Midpoint Slope ST: Midpoint Slope RT: Midpoint Slope Circumcenter Coordinates: 4) Identify the point of concurrency: Mark up the 3 congruent segments. In the figure shown, ND = 5x 1 and NE = 2x + 11. Find NF. Show all work. NF = 5) Solving a Real-Life Problem: A school has fenced in an area in the shape of a scalene triangle to use for a new playground. The school wants to place a swing set where it will be the same distance from all three fences. Should the swing set be placed at the circumcenter or the incenter of the triangular playground? Explain your reasoning. P a g e 8
6) In ΔRST, point Q is the Centroid, and SQ = 8. Find QW and SW. Show all work. QW = SW = 7) Find the coordinates of the Centroid of ΔRST with vertices R(2, 1), S(5, 8), and T(8, 3). Show all work. Hint: Use median SV to find the coordinates of the Centroid because it is easier to find a distance on a vertical segment. 8) There are three paths through a triangular park. Each path goes from the midpoint of one edge to the opposite corner. The paths meet at point P. Show all work. a. Find PS and PC when SC = 2100 feet. PS = PC = b. Find TC and BC when BT = 1000 feet. TC = BC = c. Find PA and TA when PT = 800 feet. PA = TA = What point of concurrency is shown? 9) Are the four Points of Concurrency distinct points in every triangle? Yes/No Explain you reasoning. P a g e 9
10) Tell whether the orthocenter of the triangle with the given vertices is inside, on, or outside the triangle. a. A(0, 3) B(0, 2) C(6, 3) b. J( 3, 4) K( 3, 4) L(5, 4) Inside/Outside/On Inside/Outside/On 11. Which point of concurrency is equidistant from the vertices of a triangle? What is it formed by? 12. Which point of concurrency is equidistant from the sides of a triangle? 13. What is it formed by? 14. Which point of concurrency allows you to construct the largest circle inside a triangle? 15. Which point of concurrency allows you to construct a circle that allows you to inscribe the triangle? 16. The incenter is inside the triangle. (always, sometimes, never) 17. The circumcenter is inside the triangle. 18. Where is the circumcenter of a right triangle? 19. Which point of concurrency should you find that is equidistant from three points? SKETCHING DRILL Sketch the point of concurrency listed: (label the congruent and perpendicular segments formed with appropriate marks) Incenter Circumcenter P a g e 10
Identify the point of concurrency in each diagram. (Incenter of Circumcenter) 20) Find x if N is the incenter. 21) 3 Medians are drawn. What is point c called? Find x if: ZF = 2x + 8 and KZ = 3x + 2 22) Find the altitude (h) of the isosceles triangle using the Pythagorean theorem. (a 2 + b 2 = c 2 ) Note: the altitude of an isosceles triangle bisects the base. 23) O is the circumcenter and the dashed lines are the perpendicular bisectors. Find all of the missing segment lengths: MO, PR, MN, SP, QN, MP. 24) If SU = 2, TR = 3, RV = 5, SQ = 6, PV = 12, find the lengths: PT =, PU =, VQ =, PS =, SV =, RS =, QT =. P a g e 11
Location of the Points of Concurrency (Reference Sheet) Circumcenter Incenter (always inside) Centroid (always inside) Orthocenter Acute Right Obtuse Equilateral All points are in the same place because, altitude, angle bisector, perpendicular bisector and medians are all the same. Which segments are perpendicular? Which special segments must go through the vertex always? Which special segments contain the midpoint? What are concurrent lines? Where exactly is the circumcenter of a right triangle? Where is the orthocenter of a right triangle? The slopes of perpendicular lines are: The midpoint formula is: Centroid formula is: P a g e 12