الأسبوع 12 · ميل المستقيم: زاوية الميل والمستقيمان المتوازيان والمتعامدان والنقاط على استقامة واحدة · 3 دروس
3
Perpendicular lines have slopes whose product is −1: right angles, parallelograms and rectangles
Mathematics Prep 3 Term 1, Unit 3, Lesson 6
GoalUse m₁ × m₂ = −1 to prove right angles, and rectangles.
New words
perpendicular · meeting at a right anglenegative reciprocal · turn the fraction upside down and change the signrectangle · a parallelogram with four right anglesright angle at B · the sides at B are perpendicular
Explanation
Two lines (neither vertical nor horizontal) are perpendicular exactly when m₁ × m₂ = −1. So the slope of a perpendicular line is the negative reciprocal: 2 becomes −1/2, and −2/3 becomes 3/2. For example, the line through (0, 0) and (4, 2) has slope 1/2 and the line through (4, 2) and (2, 6) has slope (6 − 2)/(2 − 4) = −2. Since (1/2) × (−2) = −1, they meet at a right angle at (4, 2). To prove a quadrilateral is a rectangle, show it is a parallelogram (opposite slopes equal) and that two adjacent sides have product −1. A horizontal line is perpendicular to a vertical line, although neither has a product to check.
Worked examples
Is the angle at B right in the triangle A(1, 1), B(5, 3), C(3, 7)? Slope AB = 2/4 = 1/2 and slope BC = (7 − 3)/(3 − 5) = −2. The product is −1, so ∠B = 90°.
Prove ABCD with A(0, 0), B(4, 2), C(2, 6), D(−2, 4) is a rectangle. AB and DC have slope 1/2; BC and AD have slope −2. So it is a parallelogram, and (1/2)(−2) = −1 gives a right angle at B. (It happens to be a square, since AB = BC = √20.)
A line has slope −2/3. A perpendicular line has slope 3/2, because (−2/3)(3/2) = −1.
Class activity
Set-square test. Draw a line with slope 1/2 on squared paper and a set-square at a point on it. Draw the perpendicular and read its slope. Do you get −2?
Practice
1
Find the slope of a line perpendicular to a line of slope 5.
−1/5.
2
Find the slope of a line perpendicular to the line through (1, 2) and (4, 0).
Slope = −2/3, so the perpendicular has 3/2.
3
Are the lines with slopes 3/4 and −4/3 perpendicular?
(3/4)(−4/3) = −1: yes.
4
Show that the triangle A(0, 0), B(2, 1), C(−1, 2) has a right angle at A.
Slope AB = 1/2, slope AC = −2, product −1.
5
The line through (−1, 2) and (3, −6) has what slope, and what is the slope of its perpendicular?
