In triangle ABC, the angle bisector of angle BAC meets line segment BC at D, such that AD = AB. Line segment AD is extended to E, such that CD = CE and angle DBE = angle BAD. Show that triangle ACE is isosceles. Please try to describe step-by-step how you achieved the answer, because i want to learn. :) Thank you for your help in advance!

DE = 11.25 units

Step-by-step explanation:

The image of the quadilateral BCED and the extensions of BD and CE to move past B and C to meet at A is presented in the attached file to this answer.

Let the angle at A be θ

Let the length of DE be x

ABC forms a triangle which van allow θ to be obtained using cosine rule.

BC² = AB² + AC² - (2×AB×AC×cos θ)

BC = 9

AB = 24

AC = 28

9² = 24² + 28² - (2×24×28×cos θ)

-1279 = -1344 cos θ

Cos θ = (1279/1344) = 0.9516

θ = 17.89°

Note that ADE also forms a triangle and we can also use cosine rule to obtain the required side DE

DE² = AD² + AE² - (2×AD×AE×cos θ)

DE = ?

AD = AB + BD = 24 + 11 = 35

AE = AC + CE = 28 + 2 = 30

θ = 17.89°

DE² = 35² + 30² - (2×35×30×cos 17.89°)

DE² = 126.54

DE = √126.54 = 11.25

Hope this Helps!!!

d: (13,-1)

step-by-step explanation:

we know the starting point is (1,5)

the ending point is (x,y)   (cristians house)

the total distance is 6 units (5: 1)  

the fast food restaurant is at (11,0) and is 5/6ths of the way from the starting point to the ending point

ending minus starting point * 5/6   is how far they went

(x-1) *5/6, (y-5) * 5/6

we then need to add the starting point to determine the location

5/6(x-1) + 1, (y-5) * 5/6 + 5

this equals the fast food location of 11,0

setting the x locations equal

5/6 (x-1) +1 =11

subtracting 1 from each side

5/6 (x-1) +1-1 =11-1

5/6(x-1) =10

multiplying by 6/5 on each side

6/5*5/6(x-1) = 10*6/5

x-1 =12

add 1 to each side

x-1+1 =12+1

x=13

do the same for the y coordinate

(y-5) * 5/6 + 5 =0

subtract 5 from each side

(y-5) * 5/6 + 5-5 =0-5

5/6 (y-5) = -5

multiplying by 6/5 on each side

6/5*5/6 (y-5) = -5 *6/5

y-5 = -6

add 5 to each side

y-5+5 = -6+5

y=-1

the ending point , or the coordinates of cristians house is

(13,-1)