10R Unit 7 Triangle Relationships CW 7.8 HW: Finish this CW 7.8 Review for Test Answers: See Teacher s Website Theorem/Definition Study Sheet! Term: Definition: Picture: Exterior Angle Theorem: Triangle Inequality: Midsegment Theorem: Median: The ext = the sum of nonadjacent interior s To have a triangle: Sum of 2 smaller sides > largest = to half the side its to connects 2 sides of a triangle at their midpoints Segment drawn from vertex to the midpoint of the opposite side (median 2 segs) Example: {2, 3, 4} x+ y = w 2+3=5 and 5 > 4 you have a triangle BM MC and Altitude: Segment drawn from vertex perpendicular to the opposite side (height of Δ) (altitude right s) Angle Bisector: bisector 2 s Perpendicular Bisector: Segment drawn to the midpoint that also forms right angles (Not always through a vertex) bisector rt s & 2 segs 2
Centroid: Point of concurrency of medians Orthocenter: Point of concurrency of altitudes Incenter: Point of concurrency of angle bisectors *right triangle = located on hypotenuse, not pictured Circumcenter: Point of concurrency of perpendicular bisectors Medians: Drawn to the MIDPOINT from opposite vertex 2:1 ratio (3 equal pieces) Point of concurrency = Centroid (always inside triangle) Altitudes: HEIGHT! (From vertex to opposite side to form a right angle) Point of concurrency = Orthocenter Could be either inside or outside Perpendicular Bisector: THE 1-2 PUNCH: Midpoint and Right s!! Point of concurrency = Circumcenter (equidistant from vertices) Not necessarily through a vertex of the triangle Angle Bisector: 2 congruent angles Point of concurrency = Incenter (always inside triangle) Equidistant from sides of the triangle 3
1. In Δ DEF, an exterior angle at F is represented by 8x + 15. If the two non-adjacent interior angles are represented by 4x + 5 and 3x + 20, find the value of x. F 8x+15 D 4x+5 3x+20 E 2. Given isosceles triangle ABC with vertex angle B, an exterior angle at B measures 22x 2, and the measure of one of the base angles is12x 3. Find the measure of the vertex angle of the triangle. 3. In which triangle do the three altitudes intersect outside the triangle? (1) a right triangle (2) an acute triangle (3) an obtuse triangle (4) an equilateral triangle 4. In Δ ABC, D is a point on AC such that BD is a median. Which statement must be true? (1) ΔABD Δ CBD (3) AD CD (2) ABD CBD (4) BD AC 5 5. Solve for x given BD = x + 3and AE = 6x + 4. 2 Assume B is the midpoint of AC and D is the midpoint of CE. 6. Find the value of x and y. 4
7. Given that AE bisects DAB. Find DE if CB = 6 and CE = 8. 8. On the set of axes below, graph and label ΔDEF with vertices at D( 4, 4 ), E( 2, 2 ), and F(8, 2 ). If G is the midpoint of EF and H is the midpoint of DF state the coordinates of G and H and label each point on your graph. Explain why GH DE. 9. Triangle JKL has vertices J(0, 0), K(4, 0) and L(0,5). Find the equation of the line for all three altitudes. 10. Point G is the incenter of Δ ACE, FG = 16, and AG = 34. Find the value of BG. 5
11. Point N is the incenter of Δ GHJ, and KH = 35, NH = 37, and NL = 2x. Find the value of x. 12. In Δ ABC, Q is the centroid and QC = 6. Find CM. Questions 13-15: In the given figure to the right, P is the centroid of ΔABC and BP = 8. 13. Find FP. 14. Find BF. 15. If AD = 3x + 7 and AB = 50, find x. 16. In the diagram of ΔABC below, Jose found centroid P by constructing the three medians. He measured CF and found it to be 6 inches. If PF = x, which equation can be used to find x? (1) x + x = 6 (2) 2x + x = 6 (3) 3x + 2x = 6 (4) 2 x + x = 6 3 17. Side PQ of ΔPQR is extended through Q to point T. Which statement is not always true? (1) m RQT > m R (2) m RQT > m P (3) m RQT = m P + m R (4) m RQT > m PQR 6
18. In the diagram of Δ ABC below, AB = 10, BC = 14, and AC = 16. Find the perimeter of the triangle formed by connecting the midpoints of the sides of Δ ABC. 19. State if a triangle is possible (yes/no) and if so, state what kind of triangle it would be (isosceles, equilateral, scalene). a.) {2, 4, 5} b.) {5, 5, 6} c.) {7, 7, 7} 20. Matching: Incenter Orthocenter Centroid Circumcenter a.) point of concurrency of the altitudes of a Δ b.) point of concurrency of the medians of a Δ c.) point of concurrency of the perpendicular bisectors of a Δ d.) point of concurrency of the angle bisectors of a Δ 21. Which of the following is false about the centroid of a triangle? (1) Its nickname is the balancing point of the triangle. (2) It is the point of concurrency of the medians of a triangle. (3) It divides each median of the triangle into two segments in the ratio of 2:1. (4) It can lie in either the interior or the exterior of the triangle. 22. Which of the following is false about the incenter of a triangle? (1) It always lies in the interior of the triangle. (2) It is equidistant from the vertices of the triangle. (3) It is the point of concurrency of the angle bisectors of the triangle. (4) It is equidistant from the sides of the triangle. 23. Prove Indirectly! Given: Δ ABC is not isosceles with vertex angle C Prove: A is not B C A B 7