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Unit 6 Lesson 1 – Part II

Connecting Algebra and Geometry

through Coordinates

Slope & Distance

By PresenterMedia.com

AKS 32: Prove simple geometric theorems algebraically using coordinates.

KEY CONCEPTS

Calculating the Distance Between Two Points

• To find the distance between two points on a

coordinate plane, you have used the Pythagorean

Theorem.

• After creating a right triangle using each point as the

end of the hypotenuse, you calculated the vertical

height (a) and the horizontal height (b).

• These lengths were then substituted into the

Pythagorean Theorem (a2 + b2 = c2) and solved for c.

• The result was the distance between the two points.

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KEY CONCEPTS

Calculating the Distance Between Two Points

• This is similar to the distance formula, which states

the distance between points (x1, y1) and (x2, y2) is

equal to (�2− �1)2+ (2− 1)2

• Using the Pythagorean Theorem, � + �� = ��, where a = (y2-y1) and b = (x2-x1), you get:

where c = the distance

KEY CONCEPTS

CALCULATING SLOPE

• To find the slope, or steepness of a line,

calculate the change in y divided by the change

in x using the formula

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KEY CONCEPTS

Parallel & Perpendicular Lines

• Parallel lines are lines that never intersect and have equal slope.

• To prove that two lines are parallel, you must show that the

slopes of both lines are equal.

• Perpendicular lines are lines that intersect at a right angle (90˚).

The slopes of perpendicular lines are always opposite

reciprocals.

• To prove that two lines are perpendicular, you must show that

the slopes of both lines are opposite reciprocals.

• When the slopes are multiplied, the result will always be –1.

• Horizontal and vertical lines are always perpendicular to each

other.

EXAMPLE 1 Calculate the distance between the points (4, 9) and (–2, 6) using

both the Pythagorean Theorem and the distance formula.

Step 1: Plot the points on a coordinate system.

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EXAMPLE 1 (cont.)Calculate the distance between the points (4, 9) and (–2, 6) using

both the Pythagorean Theorem and the distance formula.

Step 2: Draw lines to form a right triangle, using each

point as the end of the hypotenuse.

EXAMPLE 1 (cont.)Calculate the distance between the points (4, 9) and (–2, 6) using

both the Pythagorean Theorem and the distance formula.

• STEP 3: Calculate the length of the vertical side, a,

of the right triangle.

• Let (x1, y1) = (4, 9) and (x2, y2) = (–2, 6).

• |y2 – y1| = |6 – 9| = |–3| = 3

• The length of side a is 3 units.

• STEP 4: Calculate the length of the horizontal side,

b, of the right triangle.

• |x2 – x1| = |–2 – 4| = |–6| = 6

• The length of side b is 6 units.

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EXAMPLE 1 (cont.)Calculate the distance between the points (4, 9) and (–2, 6) using

both the Pythagorean Theorem and the distance formula.

• STEP 5: Use the Pythagorean Theorem to calculate the

length of the hypotenuse, c.

• a2 + b2 = c2 Pythagorean Theorem

• 32 + 62 = c2 Substitute values for a and b.

• 9 + 36 = c2 Simplify each term.

• 45 = c2 Simplify.

• √45 = √c2 Take the square root of both sides of the equation.

• c = √45 ≈ 6.7

• The distance between the points (4, 9) and (–2, 6) is √45, or approximately 6.7 units.

EXAMPLE 1 (cont.)Calculate the distance between the points (4, 9) and (–2, 6) using

both the Pythagorean Theorem and the distance formula.

• Now use the distance formula to calculate the distance between the

same points.

• Let (x1, y1) = (4, 9) and (x2, y2) = (–2, 6).

• The distance between the points (4, 9) and (–2, 6) is √45 units, or

approximately 6.7 units.

• Both calculations will produce the same results each time.

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STEPS TO DETERMINE IF TWO LINES ARE

PARALLEL or PERPENDICULAR TO EACH OTHER

Step 1: Plot the lines on a coordinate plane.

Step 2: Calculate the slope of the line through the first set

of points.

Step 3: Calculate the slope of the line through the second

set of points.

Step 4: Determine of the lines are parallel or perpendicular

to each other.

• Parallel lines have each slope

• The slope of perpendicular lines are negative reciprocals of

each other.

EXAMPLE 2Determine if the line through the points (–8, 5) and (–5, 3) is parallel

to the line through the points (1, 3) and (4, 1).

Step 1: Plot the lines on a coordinate plane.

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EXAMPLE 2 (cont.)Determine if the line through the points (–8, 5) and (–5, 3) is parallel

to the line through the points (1, 3) and (4, 1).

Step 2: Calculate the slope of the line through the first set

of points (-8, 5) and (-5, 3).

EXAMPLE 2 (cont.)Determine if the line through the points (–8, 5) and (–5, 3) is parallel

to the line through the points (1, 3) and (4, 1).

Step 3: Calculate the slope of the line through the second

set of points (1,3) and (4,1).

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EXAMPLE 2 (cont.)Determine if the line through the points (–8, 5) and (–5, 3) is parallel

to the line through the points (1, 3) and (4, 1).

Step 4: Determine of the lines are parallel or perpendicular

to each other.

EXAMPLE 3Determine if the line through the points (0, 8) and (4, 9) is

perpendicular to the line through the points (–9, 10) and (–8, 6).

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EXAMPLE 4A right triangle is defined as a triangle with 2 sides that are

perpendicular. Triangle ABC has vertices A (–4, 8), B (–1, 2), and C (7,

6). Determine if this triangle is a right triangle. When disproving a

figure, you only need to show one condition is not met.

EXAMPLE 5A square is a quadrilateral with two pairs of parallel opposite sides,

consecutive sides that are perpendicular, and all sides congruent,

meaning they have the same length. Quadrilateral ABCD has vertices

A (–1, 2), B (1, 5), C (4, 3), and D (2, 0). Determine if this quadrilateral

is a square.