In ΔABC, AB=20 cm, BC=7 cm and AC=15 cm. Side BC is produced to D such that ΔDAB∼ΔDCA. The length of CD is:
- A12 cm
- B10 cm
- C9 cm
- D8 cm
Solution & Step-by-step Explanation
Given ΔDAB∼ΔDCA.
From similarity, the ratios of corresponding sides are equal:
DC
DA
=
CA
AB
=
DA
DB
We are given AB=20 cm and AC=15 cm:
CA
AB
=
15
20
=
3
4
So,
DC
DA
=
3
4
⟹DA=
3
4
CD
And,
DA
DB
=
3
4
⟹DB=
3
4
DA
Substitute DA into the expression for DB:
DB=
3
4
(
3
4
CD)=
9
16
CD
We know that DB=BC+CD. Given BC=7 cm:
9
16
CD=7+CD
9
16
CD−CD=7
9
7
CD=7⟹CD=9 cm
From similarity, the ratios of corresponding sides are equal:
DC
DA
=
CA
AB
=
DA
DB
We are given AB=20 cm and AC=15 cm:
CA
AB
=
15
20
=
3
4
So,
DC
DA
=
3
4
⟹DA=
3
4
CD
And,
DA
DB
=
3
4
⟹DB=
3
4
DA
Substitute DA into the expression for DB:
DB=
3
4
(
3
4
CD)=
9
16
CD
We know that DB=BC+CD. Given BC=7 cm:
9
16
CD=7+CD
9
16
CD−CD=7
9
7
CD=7⟹CD=9 cm