3. Verif Properties of Triangles Since triangular frames are strong and simple to make, the are widel used to strengthen buildings and other structures. This section applies analtic geometr to verif the properties of specific triangles. Investigate How can ou verif the properties of an isosceles triangle? Tools grid paper Method 1: Use Pencil and Paper 1. raw the triangle with vertices A(, 5), B(1, ), and C(, ). Use the coordinates of the vertices to verif that ABC is isosceles. Which angles of ABC are equal?. escribe how ou could fold our drawing of ABC to confirm that it is isosceles. 3. Find the coordinates of the midpoint,, of side AB. raw the median from verte C. Use slopes to verif that this median is perpendicular to AB. A(, 5). Eplain how ou can use congruent triangles to verif that C is perpendicular to AB. escribe two different was to show that AC and BC are congruent. 0 B(1, ) C(, ) 5. Use angle sums to verif that C bisects ACB.. escribe another wa to show that C bisects ACB. 7. Reflect How are the median, the altitude, and the angle bisector at verte C related? How are these line segments related to the right bisector of AB? o ou think all isosceles triangles have these properties? Eplain our reasoning. 3. Verif Properties of Triangles MHR 117
Tools computer with The Geometer s Sketchpad Technolog Tip You can also place the vertices b choosing Plot Points from the Graph menu and tping the coordinates. Method : Use The Geometer s Sketchpad 1. Choose Show Grid from the Graph menu. Then, choose Snap Points. Move the origin so that most of the work area is in the first quadrant. Construct the triangle with vertices A(, 5), B(1, ), and C(, ).. Measure the length of each side to verif that ABC is isosceles. Which angles of ABC are equal? 3. Construct the midpoint,, of side AB. Then, construct the median from verte C. Measure AC. What can ou conclude about the median C?. Measure and compare AC and BC. 5. Turn off Snap Points. rag a verte of ABC to a new location. Move one of the other vertices until AC BC. Observe the measures of AC, AC, and BC as ou repeat this process for various locations of the vertices.. Reflect How are the median, the altitude, and the angle bisector at verte C related? How are these line segments related to the right bisector of AB? o ou think all isosceles triangles have these properties? Eplain our reasoning. Tools TI-83 Plus or TI-8 Plus graphing calculator Method 3: Use a Graphing Calculator 1. Start the Cabri Jr. application. If the aes are not displaed, choose Hide/Show from the F5 menu; then, choose Aes from the submenu. Move the origin so that most of the work area is in the first quadrant. 118 MHR Chapter 3
. raw the triangle with vertices A(, 5), B(1, ), and C(, ). Choose Coord. & Eq. from the F5 menu. Then, select the vertices to displa their coordinates. To adjust a verte, move the cursor to it, press a, and use the arrow kes to reposition the verte. 3. Choose Measure from the F5 menu, and then choose. & Length. Measure the length of each side to verif that the triangle is isosceles. Which angles of ABC are equal?. Choose Midpoint from the F3 menu and construct the midpoint,, of side AB. Then, choose Segment from the F menu and construct the median from verte C. Measure AC. What can ou conclude about the median C? 5. Measure and compare AC and BC.. rag a verte of ABC to a new location. Move one of the other vertices until AC BC. Observe the measures of AC, AC, and BC as ou repeat this process for various locations of the vertices. 7. Reflect How are the median, the altitude, and the angle bisector at verte C related? How are these line segments related to the right bisector of AB? o ou think all isosceles triangles have these properties? Eplain our reasoning. Eample 1 Centroid of a Triangle a) Verif that C(, 0) is the centroid of OPQ. b) Verif that the centroid divides each median in a :1 ratio. Q(, ) O(0, 0) C(, 0) 0 8 P(8, ) Solution a) The centroid of a triangle is the point of intersection of the three medians. Verif that C(, 0) is the centroid b showing that the coordinates of this point satisf the equations for the lines that include the three medians. 3. Verif Properties of Triangles MHR 119
Start b using the midpoint formula to find the coordinates of the midpoint, R, of side OQ. R(, ) a 1, 1 b a 0, 0 b (, ) Q(, ) R(, ) O(0, 0) C(, 0) 0 8 P(8, ) Use these coordinates to find the slope of the median from verte P(8, ). 1 m PR I could also substitute the coordinates of points 1 P and R into = m + b to get two equations () with m and b as the unknowns. Solving this 8 sstem of equations gives values for m and b. 1 Use this slope and the coordinates of point R to find the -intercept. m b 1() b b b An equation for the line that includes the median PR is. Now, substitute the coordinates of point C(, 0) into each side of the equation. L.S. R.S. 0 0 Since the coordinates of point C satisf the equation and the point lies within the triangle, C(, 0) lies on the median PR. Net, find the midpoints, S and T, of sides OP and PQ. S(, ) T(, ) a 1, 1 a 1, 1 b b a 8 a 0 8 0 (), b, b (, ) (, 0) The method used for the median PR will also show that point C lies on the medians OT and QS. However, eamining the coordinates of the points and the diagram of OPQ reveals a shortcut for the medians OT and QS. 10 MHR Chapter 3
Since S(, ), C(, 0), and Q(, ) all have the same -coordinate, these three points lie on the vertical line with equation. Therefore, point C lies on the median QS. Similarl, O(0, 0), C(, 0), and T(, 0) are collinear since these points all lie on the -ais. So, point C also lies on the median OT. R(, ) O(0, 0) C(, 0) T(, 0) 0 8 S(, ) Q(, ) P(8, ) collinear ling on the same line Since the point C(, 0) lies on all three medians, it is the centroid of OPQ. b) To find the ratio of the parts of a median on either side of the centroid, use the length formula to find the length of each part. For the median PR, compare the lengths of PC and RC. PC ( 1 ) ( 1 ) RC ( 1 ) ( 1 ) (8 ) ( 0) ( ) ( 0) () PC is twice the length of RC. To move a factor out from under a square root sign, I take the square root of the factor: n = n Since the median OT is a horizontal line segment, find the lengths of OC and TC b simpl comparing the -coordinates of the endpoints. Subtract the lesser -coordinate from the greater one. OC 1 TC 1 0 Since QS is a vertical line segment, compare -coordinates to find the lengths of QC and SC. QC 1 SC 1 0 0 () The centroid C(, 0) divides each of the medians of OPQ into a :1 ratio. 3. Verif Properties of Triangles MHR 11
Eample Midpoints of the Sides of a Triangle In ABC, is the midpoint of side AC and E is the midpoint of side BC. a) Verif that line segment E is parallel to side AB. b) Verif that line segment E is half the length of side AB. c) Use geometr software to check our calculations in parts a) and b). A(1, 3) B(5, 1) 0 C(7, 5) E 8 Solution a) First, use the coordinates of the vertices to find the coordinates of the midpoints and E. (, ) a 1, 1 b 7 a1, 3 5 b E(, ) (, ) (, 3) Now, use the coordinates of points A, B,, and E to compare the slope of AB to the slope of E. 1 1 m AB m E 1 1 1 3 3 5 1 1 1 Since the slopes are the same, AB is parallel to E. b) Use the distance formula to compare the length of E to the length of AB. E ( 1 ) ( 1 ) AB ( 1 ) ( 1 ) ( ) (3 ) (5 1) (1 3) (1) () 5 1 0 5 5 5 Therefore, E is half the length of AB. a 1, 1 b a 5 7, 1 5 b 1 MHR Chapter 3
c) Use either The Geometer s Sketchpad or Cabri Jr. to construct the triangle with vertices A(1, 3), B(5, 1) and C(7, 5). Construct the midpoint of AC and the midpoint of BC. Label these midpoints and E, respectivel. Then, construct the line segment E. Use the software s measurement tools to find the slope and length of E and of AB. Comparing these measurements shows that the slopes are equal and AB E. Therefore, E is parallel to AB and half the length of AB. Ke Concepts You can use the midpoint, length, and slope formulas to verif properties of specific triangles. Sometimes, there are several different was to verif a propert of a given triangle. Often, ou can use a shortcut for calculations involving horizontal or vertical line segments. The centroid of a triangle divides each median into two parts, with one part twice the length of the other. The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. Reasoning and Proving Representing Selecting Tools Problem Solving Connecting Reflecting Communicating Communicate Your Understanding C1 escribe two different was to verif that PQR is isosceles. 10 8 P(3, 9) R(1, ) Q(, 3) 0 8 10 1 3. Verif Properties of Triangles MHR 13
C Outline how to use analtic geometr to verif that the part of the median on one side of the centroid of STU is twice the length of the part on the other side. U(, ) S(, 1) 0 T(, ) C3 escribe how to use geometr software to verif that each line segment within VWX is parallel to one of the sides of the triangle. V(, 1) X(, ) 0 W(, 3) Practise For help with question 1, see Eample 1. 1. etermine an equation for the line shown with each triangle. a) C(, 5) A(, ) B(7, ) For help with questions and 3, see Eample.. a) Verif that E and BC are parallel. 1 A(9, 15) 1 1 10 0 8 8 E b) E(, 3) H F( 1, 0) 0 G(3, ) 0 B(3, 5) F C(13, 1) 8 10 1 1 c) J(, 3) M L(8, 3) 0 8 b) List the other line segments that are parallel. c) Verif that E BF. d) List the other line segments that have equal lengths. K(, 3) 1 MHR Chapter 3
3. Verif that PQ is twice the length of ST. Connect and Appl. a) Verif that ABC is isosceles. B(, ) b) Verif that the median from verte B is also an altitude of the triangle. 5. Use Technolog Use geometr software to verif our answers to question.. a) Find the lengths of the sides of EF. 0 Q(, ) 0 8 P(, ) C(, 8) (, ) 0 E(, ) T(, 3) A(, ) F(, 0) b) Find the slopes of the sides of the triangle. c) Classif EF. Eplain our reasoning. S(7, ) R(1, 0) 8 10 1 7. a) raw the triangle with vertices J(8, 8), K(5, 5), and L(5, 7). What tpe of triangle does JKL appear to be? b) Use analtic geometr to verif our classification of JKL. c) etermine the perimeter of the triangle. d) etermine the area of the triangle. 8. Use Technolog Use geometr software to verif our answers to question 7. 9. a) raw the triangle with vertices P(1, ), Q(, 0), and R(8, ). b) etermine the coordinates of S, the midpoint of PR, and T, the midpoint of PQ. c) Verif that ST is parallel to QR. d) Verif that ST is half the length of QR. 10. a) raw the triangle with vertices A(3, ), B(, 0), and C(5, 0). Find the midpoint of each side, and label these midpoints, E, and F. b) Verif that EF is similar to ABC. Find the ratio of the lengths of corresponding sides of these triangles. c) Verif that the area of ABC is four times the area of EF. d) How is the ratio of the lengths of corresponding sides related to the ratio of the areas of ABC and EF? 11. Use Technolog Use geometr software to verif our answers to question 10. 1. A landscape architect is drawing plans for a rigid triangular canop to provide shade in a courtard. On the drawing, the vertices of the canop are O(0, 0), P(10, 0), and Q(, 1). A single pole will support the canop. a) Verif that the triangular canop has a centroid. b) Eplain wh the centroid is a good location for attaching the canop to the support pole. 3. Verif Properties of Triangles MHR 15
13. a) Verif that the triangle with vertices J(1, ), K(3, 1), and L(0, 5) is an isosceles right triangle. b) escribe another method that ou could use to answer part a). 1. a) etermine the equations of the right bisectors of the sides of OAB. A(, ) 1. Chapter Problem a) raw a large ABC with A 3 and B C 7. Bisect B. Label the intersection of the angle bisector with side AC as point. A 3 O(0, 0) 0 B(8, 0) 8 b) etermine the coordinates of the circumcentre, the point of intersection of the right bisectors of the sides. c) What kind of triangle is OAB? Justif our answer. d) escribe the location of the circumcentre of this triangle. 15. a) Show that the right bisectors of the sides of EF all intersect at point C(, ), the circumcentre of the triangle. b) Verif that point C is equidistant from the three vertices of EF. ( 18, 1) C(, ) 0 1 1 8 0 8 1 1 8 8 1 E(, 1) F(1, ) B b) Verif that both ABC and BC are golden triangles. Eplain how ou know that these two triangles are similar. c) Construct the bisector of C, and label the intersection with B as point E. Then, bisect CE. Continue this process to produce a series of smaller and smaller golden triangles. d) raw a smooth curve through points A, B, C,, E, and so on. This curve is a golden spiral. e) Is this golden spiral a fractal? Eplain. Achievement Check 17. Verif that the points A(, 1), B(8, 5), and C(1, 1) are collinear using a) slopes b) an equation of a line c) lengths Etend 3 18. A triangle has vertices P(a, b), Q(c, d), and R(e, f ). a) etermine the coordinates of S and T, the midpoints of PQ and PR, respectivel. b) Verif that ST is parallel to QR c) Verif that ST is half the length of QR. C 1 MHR Chapter 3
19. A cevian is an line segment that joins a verte of a triangle to a point on the opposite side. In 178, the Italian mathematician Giovanni Ceva published a theorem about concurrent cevians in a triangle. Ceva s theorem states that A BE CF 1, where AE, BF, B EC FA and C are concurrent cevians in ABC. A F B E C a) raw an three concurrent cevians in a large scalene triangle. Measure the line segments to verif that Ceva s theorem applies for our triangle. b) Verif that Ceva s theorem applies for the medians of an triangle. c) Use Technolog Use geometr software to verif that Ceva s theorem applies for all triangles. Outline our method. 0. ABCE is a regular pentagon. Which of the triangles formed b the diagonals of pentagon ABCE are golden triangles? E A 1. Math Contest The golden ratio is. Show that 1.. Math Contest How man different threeletter arrangements can ou make using the letters in the word MOLLY? A T B 30 C 33 0 P E 10 S Q R 1 5 B C Makin onnections You can use The Geometer s Sketchpad to generate nested golden triangles. Construct a long line segment AB. Select point A, and choose Mark Center from the Transform menu. Select point B, and choose Rotate from the Transform menu. Change the rotation angle to 3. Construct line segments from A to B and B to B. Select points A and B, in that order. Choose Iterate from the Transform menu. Map A onto B and B onto B. Press + or - to set the number of iterations to at least 5. Click on Iterate. rawing a smooth curve through the vertices of the nested golden triangles produces a golden spiral. You can draw this spiral b hand on a printout. There are Web sites with software for drawing golden spirals. Go to www.mcgrawhill.ca/links/principles10 and follow the links to find sites that generate golden triangles and golden spirals. 3. Verif Properties of Triangles MHR 17