Straight Line

• How can the distance between two points on a coordinate grid be visualized?

  By constructing a right triangle using the differences in their x and y coordinates.
• What is the formula for calculating the distance between two points \((X1, Y1)\) and \((X2, Y2)\)?

  The distance \(D\) is given by \(D = \sqrt{(X2 - X1)^2 + (Y2 - Y1)^2}\).
• What does the distance formula extend to in three dimensions?

  It extends by adding the z-axis to the calculation.
• Given points A \((-2, 4)\) and B \((3, -1)\), how do you calculate the length of line AB?

  Calculate using the distance formula: \(D = \sqrt{(3 - (-2))^2 + (-1 - 4)^2}\).
• What type of triangle is formed if two sides are equal in length?

  An isosceles triangle.
• What is the significance of the distances calculated in the triangle PQR?

  They help determine if triangle PQR is isosceles.
• How do you verify if triangle HRC is a right angle triangle using Pythagoras?

  Check if \(D_{HR}^2 + D_{HC}^2 = D_{RC}^2\).
• What does it mean if \(a^2 + b^2 = c^2\) in the context of triangles?

  It indicates that the triangle is a right angle triangle.
• What is the midpoint formula for two coordinates \((X1, Y1)\) and \((X2, Y2)\)?

  The midpoint is given by \(\left(\frac{X1 + X2}{2}, \frac{Y1 + Y2}{2}\right)\).
• How do you find the coordinates of point B given point A and the center of a circle?

  Use the midpoint formula to solve for the coordinates of B.
• What does it mean for points to be collinear?

  It means they lie on the same straight line.
• How can you prove that points P, Q, and R are collinear?

  By showing that the gradients between pairs of points are equal.
• What is the gradient formula used to check collinearity between points P and Q?

  The gradient is calculated as \(\frac{Y2 - Y1}{X2 - X1}\).
• What must be true for points A, B, and C to be collinear?

  The gradients between A to B and B to C must be equal.
• What is the relationship between the angle of a line and its gradient?

  The gradient is equal to the tangent of the angle the line makes with the positive x-axis.
• If the angle of a line is \(32\) degrees, how do you calculate the gradient?

  The gradient is \(\tan(32^\circ)\).
• How do you find the angle of a line given two points P and Q?

  Calculate the gradient and then use the inverse tangent function.
• What is the exact value of the gradient when the angle is \(30\) degrees?

  The exact value is \(\frac{1}{\sqrt{3}}\).
• How do you find the gradient of line a b given its angle with the y-axis?

  By calculating the tangent of the angle with the x-axis.
• What are the exact values for tangent that are often used in higher maths?

  30 degrees, 45 degrees, and 60 degrees.
• How do you determine if lines A B and B C are parallel?

  By comparing their gradients.
• What is the tangent of 150 degrees related to?

  It is related to the tangent of 30 degrees.
• What is the tangent of 30 degrees?

  $\frac{1}{\sqrt{3}}$
• Why are lines A B and B C not collinear?

  Because their gradients are not equal.
• What is the relationship between the gradients of two perpendicular lines?

  The product of their gradients equals -1.
• If the gradient of one line is $\frac{1}{9}$, what is the gradient of the line perpendicular to it?


  -9
• What is the gradient of a horizontal line?

  0
• What is the gradient of a vertical line?

  Undefined
• If the gradient of line A B is 3, what is the gradient of line C D that is perpendicular to it?

  -\frac{1}{3}
• If the gradient of line A B is -4, what is the gradient of line C D that is perpendicular to it?

  \frac{1}{4}
• If the gradient of line A B is \frac{1}{2}, what is the gradient of line C D that is perpendicular to it?

  2
• If the gradient of line A B is \frac{3}{4}, what is the gradient of line C D that is perpendicular to it?

  • \frac{4}{3}
• What is the definition of an altitude in a triangle?

  It is a line drawn from one vertex to the opposite side at a right angle.
• How is the area of a triangle calculated using the altitude?

  Area = \frac{1}{2} \times \text{base} \times \text{altitude}.
• What is the relationship between the altitude and the sides of an isosceles triangle?

  The altitude bisects the opposite side.
• What is the altitude of a triangle?

  The altitude of a triangle is a line drawn from one vertex that meets the opposite side at a right angle.
• How is the area of a triangle calculated?

  The area of a triangle is calculated as half the base times the height.
• What is a vertex in the context of a triangle?

  A vertex is a corner point of the triangle.
• What happens to the altitude in an isosceles triangle?

  The altitude in an isosceles triangle cuts the base in half.
• What is a perpendicular bisector in relation to an altitude?

  A perpendicular bisector is a special type of altitude that cuts a line in half.