1-A-trig-formula-for-area-of-a-triangle.ppt

Trigonometry,Trigonometry,A Trig Formula for the Area of a Triangle,IFY Maths 1,In a,right angled,triangle,the 3 trig ratios for an angle,x,are defined as follows:,3 Trig Ratios:A reminder,opposite,hypotenuse,x,In a,right angled,triangle,the 3 trig ratios for an angle,x,are defined as follows:,hypotenuse,x,adjacent,3 Trig Ratios:A reminder,In a,right angled,triangle,the 3 trig ratios for an angle,x,are defined as follows:,opposite,x,adjacent,3 Trig Ratios:A reminder,Using the trig ratios we can find unknown angles and sides of a right angled triangle,provided that,as well as the right angle,we know the following:,either,1 side and 1 angle,or,2 sides,3 Trig Ratios:A reminder,7,y,e.g.1,e.g.2,10,8,(3 s.f.),Tip:Always start with the trig ratio,whether or not you know the angle.,3 Trig Ratios:A reminder,Scalene Triangles,We will now find a formula for the area of a triangle that is not right angled,using 2 sides and 1 angle.,a,b,and,c,are the sides opposite angles,A,B,and,C,respectively.(This is a conventional,way of labelling a triangle).,ABC is a non-right angled triangle.,A,B,C,b,a,c,Area of a Triangle,Draw the perpendicular,h,from,C,to,BA,.,N,h,C,b,a,c,A,B,Area of a Triangle,ABC is a non-right angled triangle.,Draw the perpendicular,h,from,C,to,BA,.,N,h,-,(1),In,C,b,a,c,A,B,Area of a Triangle,ABC is a non-right angled triangle.,Draw the perpendicular,h,from,C,to,BA,.,N,h,-,(1),In,C,b,a,c,A,B,Area of a Triangle,ABC is a non-right angled triangle.,h,b,a,c,c,C,N,A,B,-,(1),In,Draw the perpendicular,h,from,C,to,BA,.,Area of a Triangle,ABC is a non-right angled triangle.,h,b,a,c,a,c,C,N,B,Substituting for,h,in,(1),A,-,(1),In,Draw the perpendicular,h,from,C,to,BA,.,Area of a Triangle,ABC is a non-right angled triangle.,c,b,a,a,C,B,A,Substituting for,h,in,(1),-,(1),In,Draw the perpendicular,h,from,C,to,BA,.,Area of a Triangle,ABC is a non-right angled triangle.,Any side can be used as the base,so,Area of a Triangle,The formula always uses 2 sides and the angle formed by those sides,Area,=,=,=,Any side can be used as the base,so,Area of a Triangle,The formula always uses 2 sides and the angle formed by those sides,c,b,a,a,C,B,A,Area,=,=,=,Any side can be used as the base,so,Area of a Triangle,The formula always uses 2 sides and the angle formed by those sides,c,b,a,a,C,B,A,Area,=,=,=,Any side can be used as the base,so,Area of a Triangle,The formula always uses 2 sides and the angle formed by those sides,c,b,a,a,C,B,A,Area,=,=,=,1.Find the area of the triangle PQR.,Example,7 cm,8 cm,R,Q,P,Solution:We must use the angle formed by the 2 sides with the given lengths.,1.Find the area of the triangle PQR.,Example,7 cm,8 cm,R,Q,P,Solution:We must use the angle formed by the 2 sides with the given lengths.,We know PQ and RQ so use angle Q,1.Find the area of the triangle PQR.,Example,7 cm,8 cm,R,Q,P,Solution:We must use the angle formed by the 2 sides with the given lengths.,We know PQ and RQ so use angle Q,cm,2,(3 s.f.),A useful application of this formula occurs when we have a triangle formed by 2 radii and a chord of a circle.,Area of a Triangle,r,B,A,C,r,The area of triangle ABC is given by,SUMMARY,The area of a triangle formed by 2 radii of length,r,of a circle and the chord joining them is given by,where is the angle between the radii.,or,or,1.Find the areas of the triangles shown in the diagrams.,Exercises,radius,=4,cm.,(a),(b),X,12,cm,9,cm,B,A,C,Y,O,(a)cm,2,(3 s.f.)(b)cm,2,(3 s.f.),Ans:,