- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
The definition of the derivative practice question 8
The points , and are the vertices of a triangle with a right angle at . Find .
A right angle at is a fact about two of the triangle’s sides. Which two, and what does it say about their slopes?
The sides and meet at the right angle, so they are perpendicular and their slopes multiply to . Find the slope of , then write the slope of in terms of .
An exact value.
Theory UBC Math 100 §3.1Key results
Worked solution
Strategy
A right angle at Q makes the sides PQ and QR perpendicular, so their slopes multiply to −1. The slope of PQ is known; the slope of QR depends on k, and that condition fixes it.
Working
Rise over run from P to Q.
Perpendicular slopes multiply to −1: flip the fraction and change its sign.
Rise over run from Q to R, with k still unknown. Subtract in the same order on top and bottom.
Set the two expressions for the slope equal and solve. Check: R(8, −3) gives QR the slope −6/4 = −3/2.
Answer
The answer is k equals 8.
Check
With R = (8, −3), PQ² + QR² = 13 + 52 = 65 = PR², so Pythagoras confirms the right angle at Q.
Where this goes wrong
Flipping 2/3 to 3/2 but keeping its sign, which gives k = 0. Perpendicular slopes multiply to −1, so one of the two must be negative: QR has slope −3/2.
Leans on
Perpendicular slopes, from Functions, graphs and inverses
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