UNIT 4: Coordinate Geometry Distance, Pythagorean Theorem, Midpoint

Distance and Distance Formula Distance – the length of a line segment joining two points In the context of the coordinate plane, the distance between the ordered pairs (x 1, y 1 ) and (x 2, y 2 ) is defined by the formula on p. 171 and on your formula sheet and is illustrated on p. 171.

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

Example 2 p ) 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 ?

Example 3 page 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 )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 Two 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?

ADDITIONAL EXAMPLES P 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.

ADDITIONAL EXAMPLES P )The area of a circle is calculated using the formula A = лr 2, 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 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 a 2 + b 2 = c 2 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 PROOF P. 175

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

EXAMPLE 2 P )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?

ADDITIONAL EXAMPLES P 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.

ADDITIONAL EXAMPLES P )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.