Question

a triangle abc and bdc are similar triangles. the length of the sides of triangle abc are indicated. the area of the plane quadrilateral abdca is-----

Answer 1

Are both \(\Delta abc\) and \(\Delta bdc\) right triangles?

Answer 2

(It looks like that in your figure, but I just want to double check)

Answer 3

@ChristopherToni

Answer 4

The amusing part here is that the 3-4-5 triangle has to be a right triangle, even though it may not have been specified in the problem. The reasoning for this is that it satisfies the Pythagorean theorem: \(3^2+4^2= 9 + 16 = 25 = 5^2\).

Answer 5

ok next

Answer 6

@ChristopherToni

Answer 7

So, if \(\Delta abc\) is right, then so is \(\Delta bdc\). Using similar triangle proportions, we see that

\(\dfrac{5}{4} = \dfrac{3}{\overline{DB}}\implies 5\overline{DB} = 12 \implies \overline{DB}=\dfrac{12}{5}\)

Similarly, we see that

\(\dfrac{4}{\overline{CD}} = \dfrac{3}{12/5} \implies \dfrac{48}{5} = 3\overline{CD} \implies \overline{CD} =\dfrac{16}{5}\)

Thus, the area of the quadrilateral is the sum of the areas of the two triangles:

\(A_{\Delta abc} = \dfrac{1}{2}(4)(3) = 6\)

\(A_{\Delta bdc} = \dfrac{1}{2}\left(\dfrac{12}{5}\right)\left(\dfrac{16}{5}\right) = \dfrac{96}{25}\)

Thus, \(A_{abdc}=6+\dfrac{96}{25} = \dfrac{246}{25}\).

Does this make sense?

Answer 8

yeh thankyou:) @ChristopherToni