204 CHAPTER 2 Graphs and Functions LOOKING AHEAD TO CALCULUS In analytic geometry and calculus, the distance formula is extended to two points in space. Points in space can be represented by ordered triples. The distance between the two points 1x1, y1, z12 and 1x2, y2, z22 is given by the following expression. 21x2 - x122 + 1y 2 - y122 + 1z 2 - z122 Although our derivation of the distance formula assumed that P and R are not on a horizontal or vertical line, the result is true for any two points in a plane. EXAMPLE 2 Using the Distance Formula Find the distance between P1-8, 42 and Q13, -22. SOLUTION Use the distance formula. d1P, Q2 = 21x2 - x122 + 1y 2 - y122 Distance formula = 233 - 1-8242 + 1-2 - 422 x 1 = -8, y1 = 4, x2 = 3, y2 = -2 = 2112 + 1-622 = 2121 + 36 = 2157 S Now Try Exercise 15(a). Be careful when subtracting a negative number. A statement of the form “If p, then q” is a conditional statement. The related statement “If q, then p” is its converse. The converse of the Pythagorean theorem is also a true statement. If the sides a, b, and c of a triangle satisfy a2 +b 2 =c2, then the triangle is a right triangle with legs having lengths a and b and hypotenuse having length c. M(–2, 5) N(12, 3) Q(10, –11) x y 0 6 –10 Figure 5 EXAMPLE 3 Applying the Distance Formula Determine whether the points M1-2, 52, N112, 32, and Q110, -112 are the vertices of a right triangle. SOLUTION A triangle with the three given points as vertices, shown in Figure 5, is a right triangle if the square of the length of the longest side equals the sum of the squares of the lengths of the other two sides. Use the distance formula to find the length of each side of the triangle. d = 21x2 - x122 + 1y 2 - y122 Distance formula d1M, N2 = 2312 - 1-2242 + 13 - 522 = 2196 + 4 = 2200 d1M, Q2 = 2310 - 1-2242 + 1-11 - 522 = 2144 + 256 = 2400 = 20 d1N, Q2 = 2110 - 1222 + 1-11 - 322 = 24 + 196 = 2200 The longest side, of length 20 units, is chosen as the hypotenuse. Because A 2 200 B 2 + A 2200 B 2 = 400 = 202 is true, the triangle is a right triangle with hypotenuse joining M and Q. S Now Try Exercise 23.
RkJQdWJsaXNoZXIy NjM5ODQ=