Question
Calculate the distanet botwo \( (-5,-2) \) and \( (0,-3) \) \( (a, a+b) \) and \( (a-b, b) \)
Ask by Harris Flynn. in Zambia
Feb 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The distance between (–5, –2) and (0, –3) is √26. The distance between (a, a+b) and (a–b, b) is √(a² + b²).
Solution
To find the distance between two points, we use the distance formula:
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
1. For the points (–5, –2) and (0, –3):
x₁ = –5, y₁ = –2
x₂ = 0, y₂ = –3
x₂ – x₁ = 0 – (–5) = 5
y₂ – y₁ = –3 – (–2) = –1
Distance = √(5² + (–1)²) = √(25 + 1) = √26
2. For the points (a, a+b) and (a–b, b):
x₁ = a, y₁ = a+b
x₂ = a–b, y₂ = b
x₂ – x₁ = (a–b) – a = –b
y₂ – y₁ = b – (a+b) = –a
Distance = √[(–b)² + (–a)²] = √(b² + a²) = √(a² + b²)
So, the distances are:
• √26 for the first pair of points
• √(a² + b²) for the second pair of points.
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Simplify this solution The Deep Dive
To calculate the distance between the points \((-5,-2)\) and \((0,-3)\), you can use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Plugging in the values: \[ d = \sqrt{(0 - (-5))^2 + (-3 - (-2))^2} = \sqrt{(5)^2 + (-1)^2} = \sqrt{25 + 1} = \sqrt{26} \approx 5.1. \] Now for the distance between the points \((a, a+b)\) and \((a-b, b)\): Using the same distance formula: \[ d = \sqrt{((a - (a-b))^2 + ((a+b) - b)^2)} = \sqrt{(b)^2 + (a)^2} = \sqrt{a^2 + b^2}. \] So, you can have a distance formula to apply in various situations involving coordinates! Isn’t math intriguing?
