Question

PLEASEEEEEE HELP
Given: △ABC, D∈AC
m∠BDC=m∠ABC
AD=7, DC=9
Find: BC, BD/BA

Answer 1

To find BC and BD/BA in triangle ABC, we need to use the Law of Cosines and the angle bisector theorem.

Step-by-step explanation:

To solve for BC, we need to find the length of BC using the Law of Cosines.

Apply the Law of Cosines:

BC^2 = AD^2 + DC^2 - 2 * AD * DC * cos(BDC)

Substituting the given values:

BC^2 = 7^2 + 9^2 - 2 * 7 * 9 * cos(ABC)

Solve for BC:

BC = sqrt(7^2 + 9^2 - 2 * 7 * 9 * cos(ABC))

To find BD/BA, we can use the angle bisector theorem.

Apply the angle bisector theorem:

BD/BA = DC/AC

Substituting the given values:

BD/BA = 9/AC

Since AC is not given, we cannot calculate the exact value of BD/BA.

Answer 2

Step-by-step explanation:

Given: △ABC, D∈AC , m∠BDC=m∠ABC , AD=7, DC=9.

Solution: Let BC=x, From the given figure and the given information, it can be seen that △ABC is similar to ΔBDC by AA rule, thus using the proportionality theorem, we get

`(AC)/(BC)=(BC)/(DC)`

⇒
`(16)/(x)=(x)/(9)`

⇒
`x^(2)=144`

⇒
`x=12`

⇒BC=12

Now, calculate the similarity scale,

k=
`(AC)/(BC)`

=
`(16)/(12)=(4)/(3)`

Therefore,
`(BA)/(BD)=(4)/(3)`

⇒
`3BA=4BD`

⇒
`BA=(4)/(3)BD`