Similar Triangles

In triangle ABC, AB=20, BC=7, and CA=15. Side \overline{BC} is extended to point D so that triangle DAB is similar to triangle DCA. What is DC?
image
Source: NCTM Mathematics Teacher, November 2006

SOLUTION
image
Triangle DAB is similar to triangle DCA
\dfrac{DB}{DA}=\dfrac{DA}{DC}=\dfrac{AB}{CA}

\dfrac{DB}{DA}=\dfrac{DA}{DC}=\dfrac{4}{3}

Solving \dfrac{DA}{DC}=\dfrac{4}{3}
\dfrac{y}{x}=\dfrac{4}{3}
Cross multiply
3y=4x\qquad\qquad (1)
Solving \dfrac{DB}{DA}=\dfrac{4}{3}
\dfrac{7+x}{y}=\dfrac{4}{3}
Multiply the left hand side by 3
\dfrac{3(7+x)}{3y}=\dfrac{4}{3}
Substitute the value of 3y from Eq. (1)
\dfrac{3(7+x)}{4x}=\dfrac{4}{3}
Cross multiply
9(7+x)=16x
63+9x=16x
63=7x
x=9

Answer: 9

Alternative solution
Triangle DAB is similar to triangle DCA
\dfrac{DC}{DA}=\dfrac{AD}{BD}=\dfrac{CA}{AB}

\dfrac{DC}{DA}=\dfrac{AD}{BD}=\dfrac{3}{4}
DC=(3/4)DA\qquad\qquad (2)
BD=(4/3)AD
BD-DC=(7/12)DA
BC=(7/12)DA
7=(7/12)DA
DA=12
Substitute the value of DA into Eq. (2)
DC=(3/4)12=9.

Advertisements

About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , , , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s