UNIT 4: Coordinate Geometry Distance, Pythagorean Theorem, Midpoint

UNIT 4: Coordinate UNIT 4: Coordinate GeometryGeometry

Distance, Pythagorean Theorem, Midpoint

Distance and Distance Distance and Distance FormulaFormula

Distance – the length of a line segment joining two points

In the context of the coordinate plane, the distance between the ordered pairs (x1, y1)

and (x2, y2) is defined by the formula on p. 171 and on your formula sheet and is illustrated on p. 171.

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Examples p. 171Examples p. 171

1) A line segment has endpoints A (-3, 17) and B (15, -7). What is the length

of ?

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AB

Example 2 p. 171Example 2 p. 171

2) Four points are graphed on a coordinate plane: E (5, 5); Z (-3, -1); F (-5, -5); and K (3, 1). What is true of the length of and ?

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€

EZ

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FK

Example 3 page 172Example 3 page 172

3 An ambulance is called to a wreck on Interstate 40. Ambulance 1 is 4 miles west and 3 miles north of the wreck, and Ambulance 2 is 5 miles east and 2 miles south of the wreck. How far apart are the two ambulances from each other?

Example 4 p. 173Example 4 p. 173

4)Find the perimeter of quadrilateral ABCD with vertex locations A (0, 0); B (2, 6); C (7, 6); and D (5, 0).

ADDITIONAL EXAMPLES P. 174ADDITIONAL EXAMPLES P. 174

5Two ships are sailing on the Indian Ocean, and one calls for emergency help. The first ship calls in its coordinates from the marina as 4 miles due east and 5 miles due north of the dock. The closest ship to them is 5 miles due west and 4 miles due north of the dock. How many miles is Ship 2 from Ship 1?


6 An isosceles triangle is a triangle with at least two congruent (equal) sides.

Determine if triangle ABC with vertices A (1, 5), B (2, 2), and C (4, 1) is an isosceles triangle. Explain.


7)The area of a circle is calculated using the formula A = лr2, where л ≈ 3.14 and r = radius. What is the area of a circle with a center (7, -10) and endpoint (-10, -4)?

Pythagorean Theorem Pythagorean Theorem

the sum of the squares of the lengths of the two legs (a and b) of a right triangle is equal to the square of the length of the hypotenuse (c); another way to say this is a2 + b2 = c2

Hypotenuse – the side opposite the right angle; the longest side of a right triangle

Distance formula is another way of writing the Pythagorean Theorem. See illustration p. 175.

PROOF P. 175

EXAMPLE 1 P. 176EXAMPLE 1 P. 176

1) A line segment has endpoints G (2, 4) and H (-2, -1). What is the length of GH?

EXAMPLE 2 P. 176EXAMPLE 2 P. 176

2) Carter High School is located 3 miles west and 2 miles south of city hall, and Pizza Shack is located 1 mile east and 2 miles north of city hall. How far do students have to travel to reach Pizza Shack from Carter High School?


3 The circumference of a circle is found by using the formula C = лd, where C =

circumference, л ≈ 3.14, and d = diameter. What is the circumference of a circle with endpoints (-2, -1) and (4, -3). Assume that each unit on the grid represents 1 centimeter.


4)Triangle XYZ has vertices X (-4, 1); Y (4, 1); and Z (4, 4). Prove that the

triangle is a right triangle by using the Pythagorean Theorem and distance formula.

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UNIT 4: Coordinate Geometry Distance, Pythagorean Theorem, Midpoint