10.3 Coordinate Proof Using Distance with Segments and Triangles

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1 Name Class Date 10.3 Coordinate Proof Using Distance with Segments and Triangles Essential Question: How do ou write a coordinate proof? Resource Locker Eplore G..B...use the distance, slope,... formulas to verif geometric relationships Also G.6.B, G.6.D Deriving the Distance Formula and the Midpoint Formula Complete the following steps to derive the Distance Formula and the Midpoint Formula. To derive the Distance Formula, start with points J and K as shown in the figure. K (, ) Given: J ( 1, 1 ) and K (, ) with 1 and 1 Prove: JK = ( 1 ) + ( 1 ) J ( 1, 1 ) L Locate point L so that _ JK is the hpotenuse of right triangle JKL. What are the coordinates of L? B Find JL and LK. B the Pthagorean Theorem, J K = J L + L K. Use this to find JK. Eplain our steps. Houghton Mifflin Harcourt Publishing Compan To derive the Midpoint Formula, start with points A and B as shown in the figure. Given: A ( 1, 1 ) and B (, ) Prove: The midpoint of AB _ is M (_ 1 +,_ 1 + ). 1 B (, ) M A ( 1, 1 ) 1 What is the horizontal distance from point A to point B? What is the vertical distance from point A to point B? Module Lesson 3

2 E The horizontal and vertical distances from A to M must be half these distances. B (, ) What is the horizontal distance from point A to point M? M F What is the vertical distance from point A to point M? To find the coordinates of point M, add the distances from Step E to the and coordinates of point A and simplif. 1 1 A ( 1, 1 ) coordinate of point M: 1 + _ 1 = _ 1 + _ 1 = = _ 1 + coordinate of point M: Reflect 1. In the proof of the Distance Formula, wh do ou assume that 1 and 1?. Does the Distance Formula still appl if 1 = or 1 =? Eplain. 3. Does the Midpoint Formula still appl if 1 = or 1 =? Eplain. Houghton Mifflin Harcourt Publishing Compan Module Lesson 3

3 Eplain 1 Positioning a Triangle on the Coordinate Plane A coordinate proof is a stle of proof that uses coordinate geometr and algebra. The first step of a coordinate proof is to position the given figure in the plane. You can use an position, but some strategies can make the steps of the proof simpler. Strategies for Positioning Figures in the Coordinate Plane Use the origin as a verte, keeping the figure in Quadrant I. Center the figure at the origin. Center a side of the figure at the origin. Use one or both aes as sides of the figure. Eample 1 Write each coordinate proof. Given: B is a right angle in ABC. D is the midpoint of _ AC. Prove: The area of DBC is one half the area of ABC. Step 1 Assign coordinates to each verte. Since ou will use the Midpoint Formula to find the coordinates of D, use multiples of for the leg lengths. The coordinates of A are (0, j). The coordinates of B are (0, 0). The coordinates of C are (n, 0). Houghton Mifflin Harcourt Publishing Compan Step Position the figure on the coordinate plane. Step 3 Write a coordinate proof. ABC is a right triangle with height j and base n. area of ABC = bh = (n) (j) A (0, j) = nj square units B the Midpoint Formula, the coordinates of D = _ 0 + n (, The height of DBC is j units, and the base is n units. area of DBC = bh = (n) (j) = nj square units D B (0, 0) C (n, 0) j + 0 _ ) = (n, j). Since nj = (nj), the area of DBC is one half the area of ABC. Module Lesson 3

4 B Given: B is a right angle in ABC. D is the midpoint of _ AC. Prove: The area of ADB is one half the area of ABC. Assign coordinates and position the figure as in Eample 1A. ABC is a right triangle with height and base. area of ABC = bh = = square units B the Midpoint Formula, the coordinates of D = (, The height of ADB is units, and the base is units. area of ADB = bh = = square units A (0, j) D B (0, 0) C (n, 0) ) = (, ). Since, the area of ADB is one half the area of ABC. Reflect 4. Wh is it possible to position ABC so that two of its sides lie on the aes of the coordinate plane? Your Turn Position the given triangle on the coordinate plane. Then show that the result about areas from Eample 1 holds for the triangle. 5. A right triangle, ABC, with legs of length units and 4 units Houghton Mifflin Harcourt Publishing Compan Module Lesson 3

5 6. A right triangle, ABC, with both legs of length 8 units Eplain Proving the Triangle Midsegment Theorem In Module 8, ou learned that the Triangle Midsegment Theorem states that a midsegment of a triangle is parallel to the third side of the triangle and is half as long as the third side. You can now use a coordinate proof to show that the theorem is true. Eample Prove the Triangle Midsegment Theorem. Given: _ XY is a midsegment of PQR. Prove: _ XY _ PQ and XY = PQ Place PQR so that one verte is at the origin. For convenience, assign verte P the coordinates (a, b) and assign verte Q the vertices (c, d). Use the Midpoint Formula to find the coordinates of X and Y. The coordinates of X are X _ 0 + a (,_ 0 + b ) = X ( a,b ). The coordinates of Y are Y ( +, Find the slope of PQ and XY. slope of PQ _ = _ 1 = _ d b 1 c a = + ) = Y (, ). R (0, 0) ; slope of XY _ = _ 1 = 1 X R X P (a, b) Y P Y Q Q (c, d) Houghton Mifflin Harcourt Publishing Compan Therefore, _ PQ ǁ _ XY since. Use the Distance Formula to find PQ and XY. PQ = ( 1 ) + ( 1 ) = (c a) + (d b) = (c a) + ( d b ) = (c a) + (d b) = (c a) + (d b) = (c a) + (d b) XY = ( 1 ) + ( 1 ) = ( ) + ( ) This shows that XY = PQ. Module Lesson 3

6 Reflect 7. Discussion Wh is it more convenient to assign verte P the coordinates (a, b) and verte Q the coordinates (c, d) rather than using the coordinates (a, b) and (c, d)? Eplain 3 Proving the Concurrenc of Medians Theorem You used the Concurrenc of Medians Theorem in Module 8 and proved it in Module 9. Now ou will prove the theorem again, this time using coordinate methods. Eample 3 Prove the Concurrenc of Medians Theorem. Given: PQR with medians PL, QM, and RN _ Prove: PL, QM, and RN _ are concurrent. P N L Q M R Place PQR so that verte R is at the origin. Also, place the triangle so that point N lies on the ais. For convenience, assign point N the vertices (0, 6a). (The factor of 6 will result in easier calculations later.) N (0, 6a) Q P Since N is the midpoint of _ PQ, assign coordinates to P and Q as follows. The horizontal distance from N to P must be the same as the horizontal distance from N to Q. Let this distance be b. Then the coordinate of point P is b and the coordinate of point Q is. The vertical distance from N to P must be the same as the vertical distance from N to Q. Let this distance be c. Then the coordinate of point P is 6a c and the coordinate of point Q is. P (, ) N (0, 6a) R (0, 0) R (0, 0) Q (, ) Houghton Mifflin Harcourt Publishing Compan Complete the figure b writing the coordinates of points P and Q. Module Lesson 3

7 Now use the Midpoint Formula to find the coordinates of L and M. The midpoint of RQ is L ( +, + ) = L (, ). The midpoint of RP _ + is M (, + ) = M (, ). Complete the figure b writing the coordinates of points L and M. Q (, ) To complete the proof, write the equation of QM and use the equation to find the coordinates of point C, which is the intersection of the medians QM and _ RN. _ Then show that point C lies on PL. P (, ) N (0, 6a) C M (, ) L (, ) Write the equation of QM using pointslope form. The slope of QM ( 6a + c ) ( 3a c ) is = =. b (b) 3 Use the coordinates of point Q for the point on QM. Therefore, the equation of QM is = + ( ). R (0, 0) Since point C lies on the ais, the coordinate of point C is 0. To find the coordinate of C, substitute = 0 in the equation of QM and solve for. + Substitute = 0. = ( ) 0 Simplif the right side of the equation. = Houghton Mifflin Harcourt Publishing Compan Distributive propert = Add 6a + c to each side and simplif. = So, the coordinates of point C are C (, ). Now write the equation of PL using pointslope form. The slope of ( 6a c ) (3a + c) 3 3 PL is = =. b b 3 Use the coordinates of point P for the point on PL. Therefore, the equation of PL is = ( + ). Module Lesson 3

8 Finall, show that point C lies on PL. To do so, show that when = 0 in the equation for PL, = 4a. Substitute = 0. = Simplif right side of equation. = + Add 6a c to each side and simplif. = ( 0 + ) Reflect 8. A student claims that the averages of the coordinates and of the coordinates of the vertices of the triangle are and coordinates of the point of concurrenc, C. Does the coordinate proof of the Concurrenc of Medians Theorem support the claim? Eplain. Eplain 4 Using Triangles on the Coordinate Plane Eample 4 Write each proof. Given: A (, 3), B (5, 1), C (1, 0), D (4, 1), E (0, ), F ( 1, ) Prove: ABC DEF Step 1 Plot the points on a coordinate plane. E A Step Use the Distance Formula to find the length of each side of each triangle. AB = (5 ) + (1 3) = _ 5 = 5; BC = (1 5) + 0 (1) = _ 17 ; AC = (1 ) + (0 3) = _ 10 ; DE = (0 (4)) + ( (1)) = _ 5 = 5; D 0 C EF = (1 0) + ( ) = _ = _ 17 ; DF = = _ = _ 10 So, AB DE, BC EF, and AC DF. Therefore, ABC DEF b the SSS Triangle Congruence Theorem and ABC DEF b CPCTC. (1 (4) ) + ( (1) ) F B Houghton Mifflin Harcourt Publishing Compan Module Lesson 3

9 B Given: J (4, 1), K (0, 5), L (3, 1), M ( 1, 3), R is the midpoint of JK, _ S is the midpoint _ of LM. Prove: JSK LRM Step 1 Plot the points on a coordinate plane. 4 Step Use the Midpoint Formula to find the coordinates of R and S R ( +, + ) = R (, ) + 4 S (, + ) = S (, ) Step 3 Use the Distance Formula to find the length of each side of each triangle. JK = (0 (4) ) + (5 1) = = _ 3 KS = ( ) 0 + ( 5) = + = _ JS = ( ) (4) + ( 1 ) = + = _ LM = (1 3) + (3 1) = = _ 3 MR = ( (1) ) + ( (3) ) = + = _ LR = ( ) 3 + ( 1 ) = + = _ Houghton Mifflin Harcourt Publishing Compan So, JK, KS, and JS _. Therefore, JKS b the Reflect SSS Triangle Congruence Theorem and JSK LRM since. 9. In Part B, what other pairs of angles can ou prove to be congruent? Wh? Module Lesson 3

10 Your Turn Write each proof. 10. Given: A (4, ), B ( 3, ), C ( 1, 3), D (5, 0), E ( 1, 1), F (0, 3) Prove: BCA EFD 11. Given: P ( 3, 5), Q ( 1, 1), R (4, 5), S (, 1), M is the midpoint of PQ, _ N is the midpoint of RS. _ Prove: PQN RSM Elaborate 1. When ou write a coordinate proof, wh might ou assign p as a coordinate rather than p? 13. Essential Question CheckIn What makes a coordinate proof different from the other tpes of proofs ou have written so far? Houghton Mifflin Harcourt Publishing Compan Module Lesson 3

Evaluate: Homework and Practice 1. Eplain how to derive the Distance Formula using PQR. R Q (, ) Online Homework Hints and Help Etra Practice P ( 1, 1 ) Write each coordinate proof.

11 Evaluate: Homework and Practice 1. Eplain how to derive the Distance Formula using PQR. R Q (, ) Online Homework Hints and Help Etra Practice P ( 1, 1 ) Write each coordinate proof.. Given: B is a right angle in ABC. M is the midpoint of AC _. Prove: M is equidistant from all three vertices of ABC. Use the coordinates that have been assigned in the figure. C (0, c) B (0, 0) M A (a, 0) 3. Given: ABC is isosceles. X is the midpoint of AB, Y is the midpoint of AC, Z is the midpoint of BC _. A (a, b) Prove: XYZ is isosceles. Use the coordinates that have been assigned in the figure. X Y Houghton Mifflin Harcourt Publishing Compan B (0, 0) Z C (4a, 0) Module Lesson 3

12 4. Given: R is a right angle in PQR. A is the midpoint of PR. _ B is the midpoint of QR. Prove: AB is parallel to PQ. _ 5. Given: ABC is isosceles. M is the midpoint of AB. _ N is the midpoint of AC. _ AB AC Prove: MC NB 6. Prove the Triangle Midsegment Theorem using the figure shown here. _ Given: DE is a midsegment of ABC. Prove: DE BC and DE = BC A (q, r) D E B (0, 0) C (p, 0) 7. Critique Reasoning A student proves the Concurrenc of Medians Theorem b first assigning coordinates to the vertices of PQR as P (0, 0), Q (a, 0), and R (a, c). The student sas that this choice of coordinates makes the algebra in the proof a bit easier. Do ou agree with the student s choice of coordinates? Eplain. Houghton Mifflin Harcourt Publishing Compan Module Lesson 3

13 Write each proof. 8. Given: J (, ), K (0, 1), L ( 3, 1), P (4, ), Q (3, 4), R (1, 1) Prove: JKL PQR 9. Given: D ( 3, ), E (3, 3), F (1, 1), S (9, ), T (3, 1), U (5, 3) Prove: FDE UST 10. Given: A (, ), B (4, 4), M (, 1), N (4, 3), X is the midpoint of _ AB, Y is the midpoint of _ MN. Prove: ABY MNX Houghton Mifflin Harcourt Publishing Compan 11. Given: J ( 1, 4), K (3, 0), P (3, 6), Q ( 1, ), U is the midpoint of _ JK, V is the midpoint of _ PQ. Prove: KVJ QUP Module Lesson 3

14 Prove or disprove each statement. 1. The triangle with vertices R (, ), S (1, 4), and T (4, 5) is an equilateral triangle. 13. The triangle with vertices J (, ), K (, 3), and L ( 1, ) is an isosceles triangle. 14. The triangle with vertices A ( 1, 3), B (, 1), and C (0, ) is a scalene triangle. 15. Two container ships depart from a port at P (0, 10). The first ship travels to a location at A ( 30, 50), and the second ship travels to a location at B (70, 30). Each unit represents one nautical mile. Find the distance between the ships to the nearest nautical mile. Verif that the port is the midpoint between the two ships. 16. The support structure for a hammock includes a triangle whose vertices have coordinates G (1, 3), H (3, ), and J (1, ). a. Classif the triangle and justif our answer. b. Algebra Each unit of the coordinate plane represents one foot. To the nearest tenth of a foot, how much metal is needed to make one of the triangular parts for the support structure? Houghton Mifflin Harcourt Publishing Compan Image Credits: Dan Barnes/iStockPhoto.com Module Lesson 3

15 17. Communicate Mathematical Ideas Eplain how the perimeter of JKL compares to the perimeter of MNP. P J M K L N 18. The coordinates of the vertices of LMN are shown in the figure. Determine whether each statement is true or false. Select the correct answer for each lettered part. L (0, d) a. LMN is isosceles. True False b. One side of LMN has a length of c units. True False c. If P is the midpoint of LN, _ then OP is _ parallel to LM. _ True False d. The area of LMN is 4cd square units. True False e. The midpoint of MN is _ the origin. True False M (c, 0) 0 N (c, 0) Houghton Mifflin Harcourt Publishing Compan H.O.T. Focus on Higher Order Thinking 19. Eplain the Error A student assigns coordinates to a right triangle as shown in the figure. Then he uses the Distance Formula to show that PQ = a and RQ = a. Since PQ = RQ, the student sas he has proved that ever right triangle is isosceles. Eplain the error in the student s proof. R (a, a) P (0, 0) Q (a, 0) Module Lesson 3

0. A carpenter wants to make a triangular bracket to hold up a bookshelf. The plan for the bracket shows that the vertices of the triangle are R (, ), S (1, 4), and T (1, ).

16 0. A carpenter wants to make a triangular bracket to hold up a bookshelf. The plan for the bracket shows that the vertices of the triangle are R (, ), S (1, 4), and T (1, ). Can the carpenter conclude that the bracket is a right triangle? Eplain. 1. Analze Relationships The vertices chosen to represent an isosceles right triangle for a coordinate proof are at ( s, s), (0, s), and (0, 0). What other coordinates could be used so that the coordinate proof would be easier to complete? Eplain. Lesson Performance Task A triathlon course was mapped on a coordinate grid marked in 1kilometer units. The starting point was (0, 0). The triathlon was broken into three stages: Stage 1: Contestants swim from (0, 0) to (0.6, 0.8). Stage : Contestants biccle from the previous stopping point to (30.6, 16.8). Stage 3: Contestants run from the previous stopping point to (5.6, 8.8). The winner averaged 4 kilometers per hour for Stage 1, 50 kilometers per hour for Stage, and 13 kilometers per hour for Stage 3. What was the winner s time for the entire race? (Assume that no time elapsed between stages.) Eplain how ou found the answer. Houghton Mifflin Harcourt Publishing Compan Image Credits: Liquidlibrar/Jupiterimages/Gett Images Module Lesson 3

Skills Practice Skills Practice for Lesson 9.1

Skills Practice Skills Practice for Lesson.1 Name Date Meeting Friends The Distance Formula Vocabular Define the term in our own words. 1. Distance Formula Problem Set Archaeologists map the location of