**IBDP** **Mathematics**** Long Question Analysis Topic: Mathematics - Circumscribed Triangle**

**Exam Question: **

In Figure 5, a △ABC is drawn to circumscribe a circle of radius 3 cm, such that the segments BD and DC are respectively of lengths 6 cm and 9 cm. If the area of △ABC is 54 cm^{2}, then find the lengths of sides AB and AC.

**Answer:**

For IBDP Mathematics, you should know:

First, let's establish some "givens". We know that:

**1. OD = 3 cm****2. BD = BE = 6 cm (equal tangents)****3. DC = CF = 9 cm (equal tangents)****4. Area of triangle ABC = 54 cm**^{2 }

Now, let's start calculating:

**1. Area of triangle OBC = 1 / 2 x 15 x 3 = 45 / 2 cm**^{2}**2. Area of triangle OAC = 1 / 2 x (x + 9) x 3 = 3(x + 9) / 2 cm**^{2}**3. Area of triangle OAB = 1 / 2 x (x + 6) x 3 = 3(x + 6) / 2 cm**^{2}

54 = 3 / 2 [(x + 9) + (x + 6) + 15]

> 54 = 3 / 2 x (2x + 30)

> 54 = 3(x + 15)

> 17 = x + 15

> **x = 2**

Then, we can calculate the length of the sides as:

x + 9 = 2 + 9 = **11**; x + 6 = 2 + 6 = **8**

Altogether, the sides of triangle ABC are 11, 8 and 15 units.

Below is the completed diagram for your reference.

Work hard for your IBDP Mathematics examination!

End of analysis. Great!