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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

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Right Triangles

• A right triangle is a triangle with a right angle

hypotenuseleg

leg

egs

ypot

enus

e

3

Pythagorean Theorem

• (leg1)2 + (leg2)2 = hypotenuse2

• True only for right triangles

Leg2

Leg1

Hypotenuse

4

Pythagorean Theorem Example

• 22 + X2 = 62

• X2 = 62 – 22 • X2 = 36 – 4 = 32• X = 32• X ≅ 5.7

2

Leg1 = ?

6

Pythagorean Theorem

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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

c

b

• A triangle’s sides measure 3, 4, 6

• 62 ? 32 + 42

• 36 > 9 + 16 = 25• Obtuse

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Distance on number line

• Find AB (distance between A and B)

-3 -1-2 10-8 2-7 -5-6 -4

BA

AB = | -8 – (-5) | = | -3 | = 3

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Distance in the coordinate plane

1) On graph paper, plot A(-3, 1) and C(2, 3).

y

x

2) Draw a horizontal segment from A and a vertical segment from C.

A(-3, 1)

C(2, 3)

B(2, 1)

3) Label the intersection B and find the coordinates of B.

QUESTIONS:

What is the horizontal distance between A and B?

What is the vertical distance between B and C?

What kind of triangle is ΔABC?

If AB and BC are known, what theorem can be used to find AC?

(2 – -3) = 5

(3 – 1) = 2

right triangle

Pythagorean Theorem

What is the measure of AC? 29 ≈ 5.4

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Distance formula in the coordinate plane

212

212 yyxx

y

xA(x1, y1)

B(x2, y2)

d

then d =

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Midpoint on a number line

• The location of the midpoint is the average of the endpoints

• M =

• M = 5

4 66M = ?

2

64

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Midpoint on the coordinate plane

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Graph A(1, 1) and B(7, 9)

C

Draw AB B(7, 9)

A(1, 1)

Find the midpointof AB.

C(4, 5)

1+7 1+9,

2 2C

8 10,

2 2C

1 2 1 2,2 2

x x y yC

C = (4, 5)