8-1, 1-8 Pythagorean Theorem, Distance Formula, Midpoint Formula Objectives Apply the Pythagorean Theorem Determine whether a triangle is acute, right, or obtuse Apply the distance formula Apply the midpoint formula

Right Triangles A right triangle is a triangle with a right angle ypotenuse hypotenuse leg egs leg A right triangle is a triangle with a right angle

Pythagorean Theorem (leg1)2 + (leg2)2 = hypotenuse2 True only for right triangles

Pythagorean Theorem Example 2 Leg1 = ? 6 Pythagorean Theorem 22 + X2 = 62 X2 = 62 – 22 X2 = 36 – 4 = 32 X = 32 X ≅ 5.7

Determining Acute, Right, or Obtuse for Triangles Let a, b, c be the lengths of the sides of a triangle, where c is the longest Acute: c2 < a2 + b2 Right: c2 = a2 + b2 Obtuse: c2 > a2 + b2 A triangle’s sides measure 3, 4, 6 62 ? 32 + 42 36 > 9 + 16 = 25 Obtuse

Distance on number line -3 -1 -2 1 -8 2 -7 -5 -6 -4 Find AB (distance between A and B) AB = | -8 – (-5) | = | -3 | = 3

Distance in the coordinate plane y x C(2, 3) 1) On graph paper, plot A(-3, 1) and C(2, 3). A(-3, 1) 2) Draw a horizontal segment from A and a vertical segment from C. B(2, 1) 3) Label the intersection B and find the coordinates of B. QUESTIONS: What is the horizontal distance between A and B? (2 – -3) = 5 What is the vertical distance between B and C? (3 – 1) = 2 What kind of triangle is ΔABC? right triangle If AB and BC are known, what theorem can be used to find AC? Pythagorean Theorem What is the measure of AC? 29 ≈ 5.4

Distance formula in the coordinate plane y x A(x1, y1) B(x2, y2) d then d =

Midpoint on a number line 4 6 6 M = ? The location of the midpoint is the average of the endpoints M = M = 5

Midpoint on the coordinate plane Graph A(1, 1) and B(7, 9) y x 10 -1 2 4 6 8 -2 3 7 1 5 9 Draw AB B(7, 9) Find the midpoint of AB. C C(4, 5) A(1, 1) C = (4, 5)