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12.21 Proving a theorem with analytic geometry Using analytic geometry, prove that the diagonals of a parallelogram bisect each other.
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Given: ~ABCD, diagonals AC and BD. To Prove: AC and BD bisect each other. Plan: Use the midpoint formula to obtain the coordinates of the midpoints of the diagonals Place ~ ABCD with vertex A at the origin and side AD along the x-axis (Fig. 12-31). Then the vertices have the coordinates A(0, 0), B(a, b), C(a c, b), and D(c, 0). a c b , R, and the midpoint of BD has By the midpoint formula, the midpoint of AC has the coordinates Q 2 2 a c b , R. Then the diagonals bisect each other, since the midpoints of both diagonals are the the coordinates Q 2 2 same point.
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CHAPTER 12 Analytic Geometry
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Fig. 12-31
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12.1. State the coordinates of each lettered point in Fig. 12-32. (12.1)
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E 4 y F x G 4 2 H I 2 D
C B A x 4 L
O 2 J 4 y
Fig. 12-32
12.2. Plot each of the following points: A( 2, 3) B( 3, 2) C(0, 1) D( 3, 0) E(3, 4) F(11, 21) 2 2 G(0, 3) H(31, 0) 2 2), D(2,
(12.2)
12.3. Plot the following points: A(2, 3), B( 3, 3), C( 3, ABCD. 12.4. Plot the following points: A(4, 3), B( 1, 3), C( 3, and triangle BCD. 12.5. Find the midpoint of the segment joining (a) (0, 0) and (8, 6) (b) (0, 0) and (5, 7) (c) (0, 0) and ( 8, 12) (d) (14, 10) and (0, 0) (e) ( 20,
2). Then find the perimeter and area of square
3), D(2,
3). Then find the area of parallelogram ABCD (12.3, 12.4) (12.5)
5) and (0, 0)
(i) (3, 4) and (7, 6) (j) ( 2, 8) and ( 4, 12)
(f) (0, 4) and (0, 16) (g) (8, 0) and (0, 2) 5)
(k) (7, 9) and (3, 3) (l) (2, 1) and ( 2, 5) (12.5) 6), ( 4, 10) 1)
(h) ( 10, 0) and (0,
12.6. Find the midpoints of the sides of a triangle whose vertices are (a) (0, 0), (8, 0), (0, 6) (b) ( 6, 0), (0, 0), (0, 10) (c) (12, 0), (0, 4), (0, 0) (e) (4, 0), (0, (f) ( 1,
(d) (3, 5), (5, 7), (3, 11)
2), (0, 2), (1,
CHAPTER 12 Analytic Geometry
(12.5)
12.7. Find the midpoints of the sides of the quadrilateral whose successive vertices are (a) (0, 0), (0, 4), (2, 10), (6, 0) (b) ( 3, 5), ( 1, 9), (7, 3), (5, 1) (c) ( 2, 0), (0, 4), (6, 2), (0, (d) ( 3, 10) 8)
7), ( 1, 5), (9, 0), (5,
12.8. Find the midpoints of the diagonals of the quadrilateral whose successive vertices are (a) (0, 0), (0, 5), (4, 12), (8, 1) (b) ( 4, 1), ( 2, 3), (6, 1), (2, 8) (c) (0, 5), (0, 1), (4, 9), (4, 3)
(12.5)
12.9. Find the center of a circle if the end points of a diameter are (a) (0, 0) and ( 4, 6) (b) ( 1, 0) and ( 5, 12) (c) ( 3, 1) and (0, 5) (e) (a, b) and (3a, 5b) (f) (a, 2b) and (a, 2c)
(12.5)
(d) (0, 0) and (2a, 2b)
12.10. If M is the midpoint of AB, find the coordinates of (a) M if the coordinates of A and B are A(2, 5) and B(6, 11) (b) A if the coordinates of M and B are M(1, 3) and B(3, 6) (c) B if the coordinates of A and M are A( 2, 1) and M(2, 1)
(12.5)
12.11. The trisection points of AD are B and C. Find the coordinates of (a) B if the coordinates of A and C are A(1, 2) and C(3, 5) (b) D if the coordinates of B and C are B(0, 5) and C(12, 4) (c) A if the coordinates of B and C are B(0, 6) and C(2, 3)
(12.5)
12.12. A(0, 0), B(0, 5), C(6, 5), and D(6, 0) are the vertices of quadrilateral ABCD. (a) Prove that ABCD is a rectangle. (b) Show that the midpoints of AC and BD have the same coordinates. (c) Do the diagonals bisect each other Why 12.13. The vertices of ^ABC are A(0, 0), B(0, 4), and C(6, 0). (a) If AD is the median to BC, find the coordinates of D and the midpoint of AD. (b) If CE is the median to AB, find the coordinates of E and the midpoint of CE. (c) Do the medians, AD and CE, bisect each other Why
(12.6)
(12.6)
12.14. Find the distance between each of the following pairs of points: (a) (0, 0) and (0, 5) (b) (4, 0) and ( 2, 0) (c) (0, 3) and (0, 7) (d) ( 6, 1) and ( 6, 11) (g) ( 3, 41) 2 and ( 3, 4 1) 2
(12.8)
(e) (5, 3) and (5, 8.4) (f) ( 1.5, 7) and (6, 7)
(h) (a, b) and (2a, b)