The coordinates of the vertices of a triangle ABC are A (4 , 3) , B (7, –3) and C (0.5, p).
(a) Calculate the gradient of the line AB.
(2)
(b) Given that the line AC is perpendicular to the line AB
(i) write down the gradient of the line AC;
(ii) find the value of p
(a) To calculate the gradient (m) of the line AB, we use the formula:
m = (y2 - y1) / (x2 - x1)
Using the coordinates A (4, 3) and B (7, -3):
m = (-3 - 3) / (7 - 4)
m = -6 / 3
m = -2
The gradient of the line AB is -2.
(b) Given that AC is perpendicular to AB, we know that the product of the gradients of AC and AB is -1.
(i) The gradient of AB is -2. Let's call the gradient of AC m2.
So, we have: -2 * m2 = -1.
(ii) To find the value of p, we need to use the coordinates A (4, 3) and C (0.5, p).
Using the formula for the gradient:
m2 = (p - 3) / (0.5 - 4)
Since -2 * m2 = -1:
-2 * [(p - 3) / (0.5 - 4)] = -1
Multiplying both sides by (0.5 - 4):
-2 * (p - 3) = -1 * (0.5 - 4)
Simplifying:
-2p + 6 = 3.5 - 4
-2p + 6 = -0.5
-2p = -0.5 - 6
-2p = -6.5
Dividing both sides by -2:
p = (-6.5) / (-2)
p = 3.25
So, the value of p is 3.25.
(a) To calculate the gradient of the line AB, we can use the formula:
Gradient = (change in y-coordinates) / (change in x-coordinates)
Let's calculate it using the coordinates of points A and B:
Gradient of AB = (y-coordinate of B - y-coordinate of A) / (x-coordinate of B - x-coordinate of A)
= (-3 - 3) / (7 - 4)
= -6 / 3
= -2
Therefore, the gradient of the line AB is -2.
(b) (i) If the line AC is perpendicular to AB, it means that the product of their gradients is -1.
Using the gradient of AB, which is -2, we can write the equation:
Gradient of AC * Gradient of AB = -1
Gradient of AC * (-2) = -1
To find the gradient of AC, we rearrange the equation:
Gradient of AC = -1 / (-2)
= 1/2
Therefore, the gradient of the line AC is 1/2.
(ii) To find the value of p, we can use the coordinates of points A and C and the gradient of AC.
Using the formula:
Gradient = (change in y-coordinates) / (change in x-coordinates)
We have:
1/2 = (y-coordinate of C - y-coordinate of A) / (x-coordinate of C - x-coordinate of A)
Substituting the given coordinates of points A and C:
1/2 = (p - 3) / (0.5 - 4)
To solve for p, we can multiply both sides by the denominator:
1/2 * (0.5 - 4) = p - 3
-3/2 = p - 3
To isolate p, we add 3 to both sides:
-3/2 + 3 = p
1/2 = p
Therefore, the value of p is 1/2.