Gr 10 Math Ch 7 SUM: Analytical Geometry
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Questions and Answers

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?

    <p>The product of their gradients equals -1.</p> Signup and view all the answers

    When is the gradient of a horizontal line zero?

    <p>When there is no vertical change</p> 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?

    <p>M(x, y) = igg( rac{x₁ + x₂}{2}, rac{y₁ + y₂}{2} igg)</p> 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?

    <p>-1/3</p> Signup and view all the answers

    What type of gradient is associated with a vertical line?

    <p>Undefined</p> Signup and view all the answers

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

    <p>In ascending alphabetical order</p> 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)?

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

    How is the gradient of a line defined?

    <p>The ratio of vertical change to horizontal change between two points</p> 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?

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

    Which statement about the coordinates of a point is true?

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

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

    <p>Because squaring any number makes it positive</p> Signup and view all the answers

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

    <p>The distance between the two points</p> 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?

    <p>Add the distances AB, BC, CD, and DA</p> Signup and view all the answers

    What is the relationship between two lines that are parallel?

    <p>They are the same distance apart.</p> 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)?

    <p>(5, 5)</p> Signup and view all the answers

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

    <p>Zero</p> 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?

    <p>-0.5</p> Signup and view all the answers

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

    <p>It is undefined.</p> 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?

    <p>y = 3x + 2</p> Signup and view all the answers

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

    <p>If the gradients are equal.</p> Signup and view all the answers

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

    <p>It equals -1.</p> 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?

    <p>A quadrilateral</p> Signup and view all the answers

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

    <p>m = (y₂ - y₁) / (x₂ - x₁)</p> 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?

    <p>4/3</p> 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)?

    <p>d = , ext{sqrt} ,((x₂ - x₁)^2 + (y₂ - y₁)^2)</p> 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?

    <p>The longest side of the right triangle</p> Signup and view all the answers

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

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

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

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

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

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

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

    <p>The steepness or slope of the line connecting the points</p> 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?

    <p>8 units</p> Signup and view all the answers

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

    <p>The squared length of the hypotenuse of a right triangle</p> Signup and view all the answers

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

    <p>$d = ext{distance} = ext{sqrt}((x₁ - x₂)^2 + (y₁ - y₂)^2)$</p> 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?

    <p>It maintains a conventional approach for consistent interpretation</p> Signup and view all the answers

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

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

    Which statement correctly describes how the distance formula is derived?

    <p>It utilizes the Pythagorean theorem to connect coordinates in a right triangle.</p> 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?

    <p>In quadrant III</p> Signup and view all the answers

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

    <p>Their product is equal to negative one.</p> Signup and view all the answers

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

    <p>y = mx + c</p> Signup and view all the answers

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

    <p>It has a zero gradient.</p> 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?

    <p>$ rac{4}{6}$</p> 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?

    <p>They are equal.</p> Signup and view all the answers

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

    <p>The gradient is zero.</p> 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?

    <p>1</p> 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)?

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

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

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

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

    <p>They have equal gradients.</p> 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?

    <p>(0, -1)</p> Signup and view all the answers

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

    <p>The product of their gradients equals -1.</p> Signup and view all the answers

    For a vertical line, how is the gradient defined?

    <p>gradient equals an undefined value</p> 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?

    <p>y = 4x - 5</p> Signup and view all the answers

    Which statement accurately describes collinear points?

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

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

    <p>5</p> Signup and view all the answers

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

    <p>In alphabetical order according to the vertices</p> Signup and view all the answers

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

    <p>Theorem of Pythagoras</p> 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)?

    <p>$y₂ - y₁$</p> Signup and view all the answers

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

    <p>The ratio of vertical change to horizontal change</p> Signup and view all the answers

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

    <p>It calculates the hypotenuse of a right triangle formed by the two points</p> 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?

    <p>The differences in the x-values and the y-values of the points</p> 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)?

    <p>The product of x-coordinates</p> 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?

    <p>$ rac{4}{6}$</p> Signup and view all the answers

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