Question

Solve this problem thanks

Answer 1

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.

$\dfrac{c}{16}=\dfrac{9}{c}$

cross multiply

$c\cdot c=16\cdot9\\\\c^2=144\to c=\sqrt{144}\\\\c=12$

$\dfrac{a}{9+16}=\dfrac{16}{a}$

cross multiply

$a\cdot a=25\cdot16\\\\a^2=400\to a=\sqrt{400}\to a=20$

The length of side b we can be calculated using the Pythagorean theorem:

$b^2+20^2=(9+16)^2\\\\b^2+400=25^2\\\\b^2+400=625\ \ \ |-400\\\\b^2=225\to b=\sqrt{225}\to b=15$

Answer:

a = 20; b = 15; c = 12

Answer 2

The three missing lengths are the left hypotenuse, x, the middle altitude, y, and the right hypotenuse, z.

9/y = y/16

y^2 = 9 * 16

y^2 = 144

y = 12

9^2 + 12^2 = x^2

x^2 = 225

x = 15

12^2 + 16^2 = z^2

z^2 = 400

z = 20

From left to right, the sides measure 15, 12, and 20 units.