In a , D and E are points on the sides AB and AC respectively such that

(i) If AD = 6 cm, DB = 9 cm and AE = 8 cm, find AC.

(ii) If and AC = 15 cm, find AE.

(iii) If and AC = 18 cm, find AE.

(iv) If AD = 4, AE = 8, DB = x – 4, and EC = 3x – 19, find x.

(v) If AD = 8 cm, AB = 12 cm and AE = 12 cm, find CE.

(vi) If AD = 4 cm, DB = 4.5 cm and AE = 8 cm, find AC.

(vii) If AD = 2 cm, AB = 6 cm and AC = 9 cm, find AE.

(viii) If and EC = 2.5 cm, find AE.

(ix) If AD = x, DB = x – 2, AE = x + 2 and EC = x – 1, find the value of x.

(x) If AD = 8x - 7, DB = 5x – 3, AE = 4x - 3 and EC = (3x – 1), find the value of x.

(xi) If AD = 4x – 3, AE = 8x – 7, BD = 3x – 1 and CE = 5x - 3, find the volume x.

(xii) If AD = 2.5 cm, BD = 3.0 cm and AE = 3.75 cm, find the length of AC.

(i)

we have

DEBC

Therefore by basic proportionally theorem

AD/DB=AE/EC

6/9=8/EC

2/3=8/EC

EC=3x8/2

EC=3x4

EC=12 cm

(ii)

we have

DEBC

Therefore by basic proportionally theorem

AD/DB=AE/EC

Adding 1 both side

AD/DB +1=AE/EC +1

3/4 +1=AE+BC/BC

3+4/4=AC/EC [AE+EC=AC]

7/4= 15/EC

EC=15x4/7

EC=60/7

Now AE+EC=AC

AE+60/7=15

AE=15-60/7

AE=105-60/7

AE=45/7

AE=6.43 cm

(iii)

we have

DEBC

Therefore by basic proportionally theorem

AD/DB=AE/EC

Adding 1 both side

AD/DB +1=AE/EC +1

+1= +1

=

= AC/AE [AE+EC=AC]

5/2=18/AE

AE=

AE=36/5

AE=7.2 cm

(iv)

we have

DEBC

Therefore by basic proportionally theorem

AD/DB=AE/EC

=

4(3x-19)=8(x-4)

12x-76=8x-32

12x-8x=76-32

4x=44

x=44/4

x=11 cm

(v)

AD=8cm,AB=12cm

since BD=AB-AC

BD=12-8

BD=4 cm

DEBC

Therefore by basic proportionally theorem

AD/DB=AE/EC

8/4=12/EC

EC=

EC =6 cm

(vi)

we have

DEBC

Therefore by basic proportionally theorem

AD/DB=AE/EC

4/4.5=8/EC

EC=

EC=9cm

Now AE+EC=AC

AC=8+9

AC=17 cm

(vii)

AD=2cm, AB=6cm

Since BD=AB-AC

BD=6-2

BD=4 cm

DEBC

Therefore by basic proportionally theorem

AD/DB=AE/EC

Taking reciprocal on both side

DB/AD=EC/AE

4/2=EC/AE

Adding 1 both side

AD/DB +1=AE/EC +1

+1= +1

=

= AC/AE [AE+EC=AC]

3=9/AE

AE=

AE=3 cm

(viii) we have

DEBC

Therefore by basic proportionally theorem

AD/DB=AE/EC

4/5=AE/2.5

AE=4x2.5/5

AE=10/5

AE=2 cm

(ix) we have

DEBC

Therefore by basic proportionally theorem

AD/DB=AE/EC

=

x(x-1)=(x+2)(x-2)

x²-x=x²-2²

-x=-4

x=4 cm

(x) we have

DEBC

Therefore by basic proportionally theorem

AD/DB=AE/EC

=

(8x-7)(3x-1)=(4x-3)(5x-3)

8x(3x-1)-7(3x-1)=4x(5x-3)-3(5x-3)

24x²-8x-21x+7=20x²-12x-15x+9

24x²-20x²-29x+27x+7-9=0

4x²-2x-2=0

2[2x²-x-1]=0

2x²-x-1=0

2x²-2x-x-1=0

2x(x-1)+1(x-1)=0

(x-1)(2x+1)=0

x-1=0

x=1

or 2x+1=0

or x=-1/2

-1/2 is not possible.

So x=1

(xi) we have

DEBC

Therefore by basic proportionally theorem

AD/DB=AE/EC

=

(8x-7)(3x-1)=(4x-3)(5x-3)

24x²-8x-21x+7=20x²-12x-15x+9

24x²-20x²-29x+27x+7-9=0

4x²-2x-2=0

2[2x²-x-1]=0

2x²-x-1=0

2x²-2x-x-1=0

2x(x-1)+1(x-1)=0

(x-1)(2x+1)=0

x-1=0

x=1

or 2x+1=0

or x=-1/2

-1/2 is not possible.

So x=1

(xii) we have

DEBC

Therefore by basic proportionally theorem

AD/DB=AE/EC

2.5/3=3.75/EC

EC=3.75x3/2.5

EC=375x3/250

EC=15x3/10

EC=9/2

EC=4.5 cm

Now AC=AE+EC

AC=3.75+4.5

AC=8.25 cm