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boffeemadrid

Answer:

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

[tex]\frac{AC}{BC}=\frac{BC}{DC}[/tex]

⇒[tex]\frac{16}{x}=\frac{x}{9}[/tex]

⇒[tex]x^{2}=144[/tex]

⇒[tex]x=12[/tex]

⇒BC=12

Now, calculate the similarity scale,

k=[tex]\frac{AC}{BC}[/tex]

=[tex]\frac{16}{12}=\frac{4}{3}[/tex]

Therefore, [tex]\frac{BA}{BD}=\frac{4}{3}[/tex]

⇒[tex]3BA=4BD[/tex]

⇒[tex]BA=\frac{4}{3}BD[/tex]

MrRoyal

Similar triangles may or may not have congruent sides

The results are [tex]BC = 12[/tex] and [tex]\frac{BD }{ BA }=\frac{4}{3}[/tex]

The side lengths are given as:

  • Length AD = 7 units
  • Length DC = 9 units

Given that the triangles are similar triangles.

So, we have the following equivalent ratio

[tex]AC : BC = BC : DC[/tex]

Express as fraction

[tex]\frac{AC }{ BC }= \frac{BC }{ DC}[/tex]

Cross multiply

[tex]BC \times BC = AC \times DC[/tex]

Where:

[tex]AC = AD + DC[/tex]

[tex]AC = 7 + 9 = 16[/tex]

So, the equation becomes

[tex]BC \times BC = AC \times DC[/tex]

[tex]BC^2 = 16 \times 9[/tex]

Take square roots of both sides

[tex]BC = 4 \times 3[/tex]

[tex]BC = 12[/tex]

Also, we have the following equivalent ratio

[tex]BD : BA = BC : DC[/tex]

This gives

[tex]BD : BA = 12 :9[/tex]

Express as fraction

[tex]\frac{BD }{ BA }=\frac{ 12}{9}[/tex]

Simplify the fraction

[tex]\frac{BD }{ BA }=\frac{4}{3}[/tex]

Read more about similar triangles at:

https://brainly.com/question/14285697

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