2010 Paper 2 Q9 - coordinate geometry

The diagram shows a triangle ABC with vertices at A(0, 5), B(8, 14) and C(k, 15). Given that AB = BC,

(i) find the value of k,

A line is drawn from B to meet the x-axis at D such that AD = CD.

(ii) Find the equation of BD and the coordinates of D.

(iii) Show that the area of the triangle ABC is 2/7 of the area of the quadrilateral ABCD.

(i) Using Pythagoras’ theorem,

AB² = BC²

So,

8² + 9² = (k – 8)² + 1²

64 + 81 – 1 = (k – 8)²

144 = (k – 8)²

k – 8 = ± 12

Since k is positive, therefore,

k = 8 + 12 = 20

(ii) For AD = CD, the line BD must be perpendicular to AC.

Hence, the gradient of BD = - 20/10 = -2

Equation of BD is  y = -2x + c

At (8, 14),   14 = -16 + c

                c = 30

Hence, the equation of BD is  y = -2x + 30

For the coordinates of D, which lies on the x-axis, y = 0

Therefore  x = 15

Hence, D(15, 0)

(iii) There are several ways to go about this. One way, is to find the area of the triangle using the coordinates (remember the anti-clockwise method?) and then the area of the quadrilateral and then find their ratio. Can, but too long, and not a clever option.

The next way is to find the height of the triangle and find the area using ½ base x height. And then do the same for the quadrilateral. Again, not a clever option.

The last way is a variant of the second one. Since the base of the triangle is the same for the smaller one and the larger one (the quad is made up of 2 triangles), then, the way to calculate the ratio of the areas is simply find the ratio of BM/BD, where M is the midpoint of AC.

So, M = (10, 10)

BM = √(2² + 4²) = √(4 + 16) = √20 = 2√5

BD = √(7² + 14²) = √(49 + 196) = √245 = 7√5

Hence, BM/BD = 2√5/7√5 = 2/7  (shown)

Notice something here. I did not go about finding the actual answer for each length. The reason was because the answer, 2/7, is a nice rounded number. With the radical (√) numbers, we know we will not get a nice round number. Hence, it gave me a hint that I should keep the radical sign and I expected that they would cancel each other.