The triangles ABD, ADC and ABC are similar (AA stands for "angle, angle" and means that the triangles have two of their angles equal).
If two triangles have two of their angles equal, the triangles are similar.
Therefore the sides of triangles are in proportion.
[tex]\dfrac{c}{16}=\dfrac{9}{c}[/tex]
cross multiply
[tex]c\cdot c=16\cdot9\\\\c^2=144\to c=\sqrt{144}\\\\c=12[/tex]
[tex]\dfrac{a}{9+16}=\dfrac{16}{a}[/tex]
cross multiply
[tex]a\cdot a=25\cdot16\\\\a^2=400\to a=\sqrt{400}\to a=20[/tex]
The length of side b we can be calculated using the Pythagorean theorem:
[tex]b^2+20^2=(9+16)^2\\\\b^2+400=25^2\\\\b^2+400=625\ \ \ |-400\\\\b^2=225\to b=\sqrt{225}\to b=15[/tex]
Answer:
a = 20; b = 15; c = 12
