Gr 10 Math Ch 7 SUM: Analytical Geometry

What does the gradient of a straight line represent?

The distance between two points on the line

The slope or steepness of the line (correct)

The midpoint between two points on the line

The y-intercept of the line

What is the standard form of the equation of a straight line?

y = mx + c (correct)

y = c + mx

y = x + c

y - y₁ = m (x - x₁)

Which statement about parallel lines is correct?

They are always the same distance apart. (correct)

They are collinear.

They have different gradients.

They intersect at a right angle.

What is the condition for two lines to be perpendicular?

The product of their gradients equals -1. Signup and view all the answers

When is the gradient of a horizontal line zero?

When there is no vertical change Signup and view all the answers

How is the mid-point of the line between two points A(x₁, y₁) and B(x₂, y₂) calculated?

M(x, y) = igg( rac{x₁ + x₂}{2}, rac{y₁ + y₂}{2} igg) Signup and view all the answers

If the gradient of line WX is 3, what is the gradient of a line that is perpendicular to it?

-1/3 Signup and view all the answers

What type of gradient is associated with a vertical line?

Undefined Signup and view all the answers

What is the conventional way to name a quadrilateral based on its vertices?

In ascending alphabetical order Signup and view all the answers

What is the correct formula to calculate the distance between two points A(x₁, y₁) and B(x₂, y₂)?

$d = ext{distance} = ewline \sqrt{(x₁ - x₂)^2 + (y₁ - y₂)^2}$ Signup and view all the answers

How is the gradient of a line defined?

The ratio of vertical change to horizontal change between two points Signup and view all the answers

If points A(1, 1) and B(3, 4) are used to calculate the gradient, what is the formula applied?

Gradient = $rac{y₂ - y₁}{x₂ - x₁}$ Signup and view all the answers

Which statement about the coordinates of a point is true?

A point is an ordered pair of numbers (x, y). Signup and view all the answers

In the calculation of the distance formula, why is it valid to say that $(x₁ - x₂)^2 = (x₂ - x₁)^2$?

Because squaring any number makes it positive Signup and view all the answers

What does the hypotenuse of the triangle formed by points A(x₁, y₁) and B(x₂, y₂) represent?

The distance between the two points Signup and view all the answers

For quadrilateral ABCD, if the coordinates are A(1, 1), B(3, 1), C(3, 3), and D(1, 3), which sequence describes its perimeter calculation?

Add the distances AB, BC, CD, and DA Signup and view all the answers

What is the relationship between two lines that are parallel?

They are the same distance apart. Signup and view all the answers

What does the mid-point formula yield for the coordinates of M when given points A(2, 4) and B(6, 8)?

(5, 5) Signup and view all the answers

What is the gradient of a line that runs parallel to the x-axis?

Zero Signup and view all the answers

If line AB has a gradient of 2, what is the gradient of a line that intersects it perpendicularly?

-0.5 Signup and view all the answers

Which of the following accurately describes the gradient of a vertical line?

It is undefined. Signup and view all the answers

Using the formula for a straight line, what will the equation be if point (1, 2) is used and the gradient is 3?

y = 3x + 2 Signup and view all the answers

How can you determine if two points are collinear using gradients?

If the gradients are equal. Signup and view all the answers

What is true regarding the product of gradients for two lines intersecting at a right angle?

It equals -1. Signup and view all the answers

What do the coordinates A(1, 1), B(3, 1), C(3, 3), and D(1, 3) form on the Cartesian plane?

A quadrilateral Signup and view all the answers

How is the gradient of a line calculated based on points A(x₁, y₁) and B(x₂, y₂)?

m = (y₂ - y₁) / (x₂ - x₁) Signup and view all the answers

Given the points A(2, 3) and B(5, 7), what is the gradient of the line joining these points?

4/3 Signup and view all the answers

Which of the following represents the correct distance formula between two points A(x₁, y₁) and B(x₂, y₂)?

d = , ext{sqrt} ,((x₂ - x₁)^2 + (y₂ - y₁)^2) Signup and view all the answers

In the context of the distance formula, what is the geometric significance of the hypotenuse formed by points A and B?

The longest side of the right triangle Signup and view all the answers

When plotting the quadrilateral ABCD, which sequence of points would correctly represent the figure?

A(1, 1) to B(3, 1) to D(1, 3) to C(3, 3) Signup and view all the answers

What is the role of the theorem of Pythagoras in determining the distance between two points?

It helps in deriving the formula for the distance using properties of right triangles. Signup and view all the answers

Which statement accurately describes the coordinates of a point on the Cartesian plane?

They represent the horizontal and vertical positions in a two-dimensional space. Signup and view all the answers

What does the gradient of a line specifically indicate about the relationship between two points?

The steepness or slope of the line connecting the points Signup and view all the answers

If a quadrilateral is formed by points A(1, 1), B(3, 1), C(3, 3), and D(1, 3), what is the perimeter of this quadrilateral?

8 units Signup and view all the answers

In the distance formula, what does the expression $(x₁ - x₂)^2 + (y₁ - y₂)^2$ represent geometrically?

The squared length of the hypotenuse of a right triangle Signup and view all the answers

Which formula would you use to find the distance between points A(2, 5) and B(6, 9)?

$d = ext{distance} = ext{sqrt}((x₁ - x₂)^2 + (y₁ - y₂)^2)$ Signup and view all the answers

When determining the order of points for drawing a quadrilateral, why is it significant to write them in alphabetical order?

It maintains a conventional approach for consistent interpretation Signup and view all the answers

How is the gradient calculated for two points A(1, 2) and B(4, 6)?

$m = rac{y₂ - y₁}{x₂ - x₁}$ Signup and view all the answers

Which statement correctly describes how the distance formula is derived?

It utilizes the Pythagorean theorem to connect coordinates in a right triangle. Signup and view all the answers

If the coordinates of point A are negative, such as A(-3, -2), how are these coordinates represented on the Cartesian plane?

In quadrant III Signup and view all the answers

What can be concluded about the gradients of two perpendicular lines?

Their product is equal to negative one. Signup and view all the answers

Which equation correctly represents the relationship between the y-intercept and the gradient of a straight line?

y = mx + c Signup and view all the answers

What is the primary characteristic of a horizontal line in terms of gradient?

It has a zero gradient. Signup and view all the answers

If the coordinates of points A and B are A(2, 3) and B(8, 7), what is the gradient of the line segment joining them?

$rac{4}{6}$ Signup and view all the answers

How do the mid-points of two segments A(1, 2) to B(3, 4) and C(5, 6) to D(7, 8) relate to each other?

They are equal. Signup and view all the answers

For points A(1, 3) and B(4, 3), which statement regarding the gradient is accurate?

The gradient is zero. Signup and view all the answers

If the coordinates of two points are A(0, 0) and B(5, 5), using the gradient formula, what will the gradient of line AB be?

1 Signup and view all the answers

What is the formula used to define the mid-point coordinates between any two points A(x₁, y₁) and B(x₂, y₂)?

M(x, y) = igg(rac{x₁ + x₂}{2}, rac{y₁ + y₂}{2}igg) Signup and view all the answers

Which formula correctly computes the gradient of the line through two points (x₁, y₁) and (x₂, y₂)?

$m = rac{y₂ - y₁}{x₂ - x₁}$ Signup and view all the answers

What is the relationship between the gradients of two lines, if they are parallel?

They have equal gradients. Signup and view all the answers

Trying to find the coordinates of mid-point M of line segment AB. If A(-2, 4) and B(2, -6), what are the coordinates of M?

(0, -1) Signup and view all the answers

If two lines are perpendicular, what can be said about their gradients?

The product of their gradients equals -1. Signup and view all the answers

For a vertical line, how is the gradient defined?

gradient equals an undefined value Signup and view all the answers

What is the equation of a straight line that passes through the points (3, 7) with a gradient of 4?

y = 4x - 5 Signup and view all the answers

Which statement accurately describes collinear points?

They have equal gradients between any two of them. Signup and view all the answers

What is the y-intercept of the line represented by the equation y = 3x + 5?

5 Signup and view all the answers

What should the order of the points be when naming a quadrilateral based on its vertices?

In alphabetical order according to the vertices Signup and view all the answers

In deriving the distance formula, which theorem is primarily utilized?

Theorem of Pythagoras Signup and view all the answers

Which of the following expressions accurately represents the vertical change in a line segment between two points A(x₁, y₁) and B(x₂, y₂)?

$y₂ - y₁$ Signup and view all the answers

What is the definition of the gradient when considering two points on a Cartesian plane?

The ratio of vertical change to horizontal change Signup and view all the answers

Which of the following correctly describes the geometric significance of the distance formula on the Cartesian plane?

It calculates the hypotenuse of a right triangle formed by the two points Signup and view all the answers

When determining the gradient of a line segment between two points A(x₁, y₁) and B(x₂, y₂), what values are compared?

The differences in the x-values and the y-values of the points Signup and view all the answers

Which expression does not contribute to the calculation in the distance formula derived from points A(x₁, y₁) and B(x₂, y₂)?

The product of x-coordinates Signup and view all the answers

Given points A(2, 3) and B(8, 7), what is the gradient of the line segment connecting these points?