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2.1 Determine whether the triangle with vertices A( 1, 2), B(4, 7), C( 3, 6) is isosceles.
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25 + 25 = 50 [ 1 ( 3)]2 + (2 6)2 = (2)2 + ( 4)2 = 4 + 16 = 20 [4 ( 3)]2 + (7 6)2 = 72 + 12 = 49 + 1 = 50 ( 1 4)2 + (2 7)2 = ( 5)2 + ( 5)2 =
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AB = AC = BC =
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Since AB = BC, the triangle is isosceles.
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2.2 Determine whether the triangle with vertices A( 5, 3), B( 7, 3), C(2, 6) is a right triangle.
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Use (2.1) to nd the squares of the sides, AB = ( 5 + 7)2 + ( 3 3)2 = 22 + ( 6)2 = 4 + 36 = 40 BC = ( 7 2)2 + (3 6)2 = 81 + 9 = 90 AC = ( 5 2)2 + ( 3 6)2 = 49 + 81 = 130 Since AB + BC = AC ,
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ABC is a right triangle with right angle at B.
2 2 2
geometry angle.
The converse of the Pythagorean theorem is also true: If AC = AB + BC in
ABC, then ABC is a right
2.3 Prove by use of coordinates that the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.
Let the origin of a coordinate system be located at the right angle C; let the positive x-axis contain leg CA and the positive y-axis leg CB [see Fig. 2-7(a)]. Vertex A has coordinates (b, 0), where b = CA; and vertex B has coordinates (0, a), where a = BC. Let M be the midpoint of the hypotenuse. By the midpoint formulas (2.2), the coordinates of M are (b/2, a/2).
Fig. 2-7
Now by the Pythagorean theorem, MA = MB = AB = 2 a 2 + b2 2
CHAP. 2]
COORDINATE SYSTEMS IN A PLANE
and by the distance formula (2.1),
2 2 a b 0 + 0 = 2 2
MC = For any positive numbers u, v,
a2 b2 + = 4 4
a 2 + b2 4
algebra
u v= v
u v = u v a 2 + b2 = 4
and so
u u = v v
a 2 + b2 2
Hence, MA = MC. [For a simpler, geometrical proof, see Fig. 2-7(b); MD and BC are parallel.]
Supplementary Problems
2.4 In Fig. 2-8, nd the coordinates of points A, B, C, D, E, and F.
Fig. 2-8
2.5 Draw a coordinate system and mark the points having the following coordinates: (1, 1), (4, 4), ( 2, 2), (3, 3), (0, 2), (2, 0), ( 4, 1). 2.6 Find the distance between the points: (a) (2, 3) and (2, 8); (b) (3, 1) and (3, 4); (c) (4, 1) and (2, 1); (d) ( 3, 4) and (5, 4).
2.7 Draw the triangle with vertices A(4, 7), B(4, 3), and C( 1, 7) and nd its area. 2.8 If ( 2, 2), ( 2, 4), and (3, 2) are three vertices of a rectangle, nd the fourth vertex. 2.9 If the points (3, 1) and ( 1, 0) are opposite vertices of a rectangle whose sides are parallel to the coordinate axes, nd the other two vertices. 2.10 If (2, 1), (5, 1), and (3, 2) are three vertices of a parallelogram, what are the possible locations of the fourth vertex 2.11 Give the coordinates of all points on the line passing through the point (2, 4) and parallel to the y-axis.
COORDINATE SYSTEMS IN A PLANE
[CHAP. 2
2.12 Find the distance between the points:
(a) (2, 6) and (7, 3); (b) (3, 1) and (0, 2); (c) (4, 1 ) and ( 1 , 3). 2 4
2.13 Determine whether the three given points are vertices of an isosceles triangle or of a right triangle (or of both). Find the area of each right triangle. (a) ( 1, 2), (3, 2), (7, 6) (b) (4, 1), (1, 2), (3, 8) (c) (4, 1), (1, 4), ( 4, 1)
2.14 Find the value of k such that (3, k) is equidistant from (1, 2) and (6, 7). 2.15 (a) Are the three points A(1, 0), B( 7 , 4) and C(7, 8) collinear (that is, all on the same line) [Hint: If A, B, C form a triangle, 2 the sum of two sides, AB+BC must be greater than the third side, AC. If B lies between A and C on a line, AB+BC = AC.] (b) Are the three points A( 5, 7), B(0, 1), and C(10, 11) collinear 2.16 Find the midpoints of the line segments with the following endpoints: and (5, 3). (a) (1, 1) and (7, 5); (b) ( 3 , 4) and (1,0); (c) ( 2, 1) 2
2.17 Find the point (a, b) such that (3, 5) is the midpoint of the line segment connecting (a, b) and (1, 2). 2.18 Prove by use of coordinates that the line segment joining the midpoints of two sides of a triangle is one-half the length of the third side. 2.19 Prove by use of coordinates that the line segments joining the midpoints of opposite sides of a quadrilateral ABCD bisect each other.
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