3.2

Gradient and length

9 flashcards to master Gradient and length

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Key Concept Flip

What is the formula to calculate the gradient (m) of a line given two points (x₁, y₁) and (x₂, y₂)?

Answer Flip

The gradient, often denoted as 'm', is calculated as the change in y divided by the change in x. The formula is: m = (y₂ - y₁) / (x₂ - x₁).

Example: points (1, 2) and (3, 6) have a gradient of (6-2)/(3-1) = 2.
Key Concept Flip

Explain the meaning of 'rise over run' in the context of gradient.

Answer Flip

'Rise over run' is a visual representation of gradient. 'Rise' refers to the vertical change (change in y), and 'run' refers to the horizontal change (change in x). The gradient is the ratio of the rise to the run.

Key Concept Flip

Calculate the length of the line segment connecting points A(2, 3) and B(5, 7).

Answer Flip

Use the distance formula (Pythagoras): √[(x₂ - x₁)² + (y₂ - y₁)²]. So, length AB = √[(5-2)² + (7-3)²] = √(9 + 16) = √25 = 5.

Key Concept Flip

Describe how the Pythagorean theorem relates to finding the distance between two points on a coordinate plane.

Answer Flip

The distance between two points is the hypotenuse of a right-angled triangle. The legs of the triangle are the difference in x-coordinates and the difference in y-coordinates. Thus, a² + b² = c² (distance squared).

Key Concept Flip

Find the midpoint of the line segment joining the points (1, 4) and (5, 2).

Answer Flip

The midpoint is found by averaging the x-coordinates and averaging the y-coordinates. Midpoint = [(x₁ + x₂) / 2, (y₁ + y₂) / 2]. Therefore, the midpoint is [(1+5)/2, (4+2)/2] = (3, 3).

Key Concept Flip

Explain what a negative gradient indicates about the slope of a line.

Answer Flip

A negative gradient indicates that the line slopes downwards from left to right. As the x-value increases, the y-value decreases.

Example: y = -2x + 3 has a negative gradient.
Key Concept Flip

A line segment has a length of 10 units and one endpoint at (1, 1). If the x-coordinate of the other endpoint is 7, what are the possible y-coordinates?

Answer Flip

Using the distance formula: 10 = √[(7-1)² + (y-1)²]. Solving for y: 100 = 36 + (y-1)², so (y-1)² = 64. Thus y-1 = ±8, giving y = 9 or y = -7.

Key Concept Flip

What does a gradient of zero indicate about a line?

Answer Flip

A gradient of zero indicates that the line is horizontal. The y-value remains constant, regardless of the x-value.

Example: the line y = 5 has a gradient of zero.
Key Concept Flip

If the gradient of a line is undefined, describe the orientation of the line.

Answer Flip

An undefined gradient means the line is vertical. This occurs when the change in x is zero (division by zero). The x-value remains constant, regardless of the y-value.

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3.1 Coordinates 3.3 Equations of lines

About Gradient and length (3.2)

These 9 flashcards cover everything you need to know about Gradient and length for your Cambridge IGCSE Mathematics (0580) exam. Each card is designed based on the official syllabus requirements.

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