Question

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Point A is the point of concurrency of the angle bisectors of ΔDEF.


What is the length of ZA?

Answer 1

I believe 3cm.

Does it have any options? you may have to round up or down.

Answer 2

Answer:

ZA = 3cm

Step-by-step explanation:

First of all we are going to find the length of YA. Since the triangle YAD is a right triangle we simple use the Pythagorean theorem. Which state the the square of the  length of  the hypotenuse (the side oposite to the right angle )is equal to the sum of the squares of the other two sides. This is:

$c^{2}=a^{2} + b^{2}$

where c is the  hypotenuse.

With this we can find the length of YA as follows:

$5^{2} = 4^{2}+\bar{YA}^{2}$

$5^{2} -4^{2}=\bar{YA}^{2}$

$\bar{YA}^{2}=9$

$\bar{YA}=3$

Thus, $\bar{YA} = 3cm$

Now using YA length and sine function for the right triangle,

$sin(\theta)= \dfrac{oposite side}{hypotenuse}$,

if we set $\theta$ to be the angle YFA this leads us to

$sin (\angle{YFA{) =\dfrac{3}{6}$

$\angle{YFA}=arcsin(\dfrac{3}{6})$

$\angle{YFA} =30°$

So, the angle YFA is iqual to 30°.

Since FA is separating the angle YFZ, the angle YFA is iqual to the angle AFZ (this is because point A  is the point of concurrency of the angle bisectors of ΔDEF)

If we call the sine function using the angle AFZ, we get :

$sin(\angle{AFZ}) =\dfrac{\bar{ZA}}{6}$

$\bar{ZA} = 6 sin(\angle{AFZ})$

$\bar{ZA} = 6 sin(30°)$

$\bar{ZA} =3$

So, length of ZA is 3 cm.