In a right triangle ΔABC, the length of leg AC = 5 ft and the hypotenuse AB = 13 ft. Find: Chapter Reference b The length of the angle bisector of angle ∠A.
Accepted Solution
A:
Answer:The length of the angle bisector of angle ∠A is 6.01.Step-by-step explanation:It is given that length of leg AC = 5 ft and the hypotenuse AB = 13 ft.Using pythagoras theorem[tex](AB)^2=(BC)^2+(AC)^2[/tex][tex](13)^2=(BC)^2+(5)^2[/tex][tex]169=(BC)^2+25[/tex][tex]BC=12[/tex][tex]\sin A=\frac{\text{perpendicular}}{\text{hypotenuse}}[/tex][tex]\sin A=\frac{BC}{AB}[/tex][tex]A=\sin ^{-1}\frac{12}{13}[/tex][tex]A=67.38[/tex]Bisector divides the angle in two equal parts, therefore,[tex]A'=\frac{67.38}{2} =33.69[/tex]In triangle ACD.[tex]\cos A'=\frac{\text{Base}}{\text{Hypotenuse}}[/tex][tex]\cos A'=\frac{AC}{AD}[/tex][tex]\cos (33.69^{\circ})=\frac{5}{AD}[/tex][tex]0.832=\frac{5}{AD}[/tex][tex]AD=\frac{5}{0.832} =6.009\approx 6.01[/tex]Therefore the length of the angle bisector of angle ∠A is 6.01.