1. 9.2 The Area of a Triangle Objective To find the area of a triangle given the lengths of two sides and the measure of the included angle.

2. The Area of a Triangle When the lengths of two sides of a triangle and the measure of the included angle are known, the triangle is uniquely determined. Therefore, the area of the triangle is unique. As we know, the area of a triangle, denoted as K, is By the right triangle trigonometry, we know that So Then the area of a triangle is

3. If some other pair of sides and the included angle of  ABC were known, we could repeat the procedure for finding the area and thereby obtain two other area formulas. The area of  ABC is given by: The Area of a Triangle

4. Example 1: Two sides of a triangle have lengths 7 cm and 4 cm. The angle between the sides measures 73º. Find the area of the triangle. [Solution]

5. Example 2: The area of  PQR is 15. If p = 5 and q = 10, find all possible measures of  R.  R can be acute or obtuse. From this example, we can see that the converse of the following statement is NOT TRUE! When the lengths of two sides of a triangle and the measure of the included angle are known, the triangle is uniquely determined. Therefore, the area of the triangle is unique.

6. A segment of a circle is the region bounded by an arc of the circle and the chord connecting the endpoints of the arc. What is the length of the chord AB? A B O C Draw line OC  chord AB at C, then OC is the bisector of chord AB and  AOB. In the right  AOC, The length of AC is Then the length of chord AB is And the area of  AOB is

7. Practice: Express the area of  AOB in a different way. After we learn the double angle formula, we will know the two formulas are the same. A B O C Then the length of chord AB is And the area of  AOB is And OC, the height of  AOB, is

8. We have already known that the area of a sector is A B O C What is the area of a segment? Obviously, the area of the segment is the difference of the area of the sector and that of central  AOB.

12. Example 4: Given  ABC with an inscribed circle as shown below. Show that the radius r of the circle is given by In addition, find the radius of the inscribed circle if a = 14, b = 12, and c = 8. (Hint: Heron’s formula) r r r B A O C [Proof] The area of  ABC, K  ABC, is the sum of the area of  AOB,  BOC, and  AOC. Since the radii of inscribed circle are perpendicular to all three sides. So the areas of  AOB,  BOC, and  AOC are respectively

13. Example 4: Given  ABC with an inscribed circle as shown below. Show that the radius r of the circle is given by In addition, find the radius of the inscribed circle if a = 14, b = 12, and c = 8. (Hint: Heron’s formula) r r r B A O C [Solution] The area of  ABC, K  ABC, can be calculated by Heron’s formula: We then calculate the perimeter (a + b + c) = 14 + 12 + 8 = 36, so s = 18. Thus the area of  ABC is

14. Example 5: Given  ABC with an inscribed circle as shown below. Show that the radius r of the circle is given by (Hint: Heron’s formula) r r r B A O C [Proof] Since s = (a + b + c)/2, then a + b + c = 2s. By applying the conclusion and Heron’s formula from the last example, we have

15. Assignment P. 342 #1 – 13, 19, 21