Question

In the diagram, the points A and C lie on the x- and y-axes respectively and the equation of AC is 2y + x = 16.?

The point B has coordinates (2, 2). The perpendicular from B to AC meets AC at the point X.

(i) Find the coordinates of X.
The point D is such that the quadrilateral ABCD has AC as a line of symmetry.

(ii) Find the coordinates of D.
(iii) Find, correct to 1 decimal place, the perimeter of ABCD

Answer 1

(1)

Step 1:

The equation of AC  is .

Rewrite the equation in the slope -intercept  form : .

Subtract x on each side.

Divide each side by 2.

Compare the above equation with standard form.

Slope and y - intercept is 8.

Step 2:

The perpendicular from point B to AC meets AC at the point X.

The slope of the perpendicular to line AC is .

Point slope form of equation is .

Substitute slope and point .

A line perpendicular to AC is .

Step 3:

Find the point of intersection of line  and its perpendicular  .

Substitute x = 4 in .

The point of intersection line AC and the perpendicular from B to AC is X = (4, 6).

Solution:

The coordinates of X  is (4, 6).

Answer 2

(2)

Step 1:

The point X is the mid point of BD . (Since line AC is symmetry) 

Mid point .

Substitute X (4, 6) and .

Let the point D (a, b).

So the point D is (6, 10).

Solution:

The point D is (6, 10).

Answer 3

(3)

Step 1:

The equation of AC is .

The point A is on the x - axis => so substitute y = 0 in AC, to find the point A.

The point A is (16, 0).

The point C is on the y - axis => so substitute x = 0 in AC ,to find the point C.

The point C is (0, 8).

Step 2:

The perimeter of the quadrilateral is P = sum of all sides.

The given quadrilateral is look like kite.

It has 2 set of equal length sides.

BC = CD and AB = AD.

Perimeter  P = BC + CD + AB + AD.

P = 2BC + 2AB .

Step 3:

Length of BC.

Length of the two points   is .

Substitute and .

Length of AB.

Length of the two points   is .

Substitute and .

Perimeter  :

Solution:

The perimeter is 40.93 units.