In the diagram, A(5 ; 3), B(0 ; 1/2), C and E(6 ; -4) are the vertices of a trapezium having BA \|\| CE - NSC Mathematics - Question 3 - 2022 - Paper 2

First, we need to find the gradient of the line CE. Using the coordinates for C (to be calculated) and E(6, -4), the formula is:

1. Find the coordinates of C: We calculate it using the property that BA is parallel to CE.
2. Write the coordinates for both points and find m:

Using a placeholder for the coordinates of C:

• Let C be (x_c, y_c).
• Use the gradient formula like before to find m for CE.

Then write the equation of the line in the form (y = mx + c). Using the coordinates of point E:

$y + 4 = m(x - 6)$

Rearranging gives us:

$y = mx - 6m - 4$.

Given that BA || CE, we know they share the same gradient. As we've computed the gradient of AB as (\frac{1}{2}), the coordinates of C can be established by considering the properties of trapeziums. Let us apply these relationships: This usually requires additional geometric understanding or coordinate deduction, so,

Assuming based on similarity:

• C: (6, y_c) where applying conditions from points B and E helps calculate final coordinates.

To find the area of quadrilateral ABCD, one effective method is to divide it into two triangles (ABD and BCD) and calculate their respective areas:

1. Area of triangle ABD:

   • Using the base A(5, 3) and height determined by the y-coordinate of B(0, 1/2).
   • Area = (\frac{1}{2} \times \text{base} \times \text{height})

2. Area of triangle BCD:

   • Similarly apply the formula with vertices defined for B, C, and D.

Combine these two areas to get the area of quadrilateral ABCD.

The reflection of point E(6, -4) in the y-axis is K(-6, -4). The coordinates of K are thus:

K: (-6, -4).

To determine the perimeter of triangle ΔKEC, we must calculate the length of each side (KE, EC, and KC) using the distance formula:

For points K(-6, -4), E(6, -4), and C(x_c, y_c):

1. Distance KE:

   • Horizontal distance between K and E: (KE = | -6 - 6 | = 12)

2. Distance EC:

   • Calculate as appropriate based on C's coordinates.

3. Distance KC:

   • Similar calculation based on K and C.

Finally, sum these distances for the perimeter.

To find the angle ∠KCE, we will apply trigonometric ratios:

1. Use the known lengths of KE and CE to apply:

$\tan(\angle KCE) = \frac{\text{opposite}}{\text{adjacent}} = \frac{KE}{CE}$

2. Substitute into the equation and solve for (\angle KCE). Use a calculator for trigonometric functions as necessary to find the precise angle measure.