Note: The question can be rewritten as follows (to make it more reader friendly):
What is the value of x? Enter your answer, as a decimal, in the box.
(a) Triangle V T K with segment T Y such that Y is on segment V K, between V and K. Angle V T Y is congruent to angle Y T K. V T equals 95.2 centimeters, V Y equals 34 centimeters, V K equals x, and T K equals 168 centimeters.
(b) Triangle M N P with segment A B parallel to segment N P and A is between M and N and B is between M and P. M N equals 67.2 meters, A N equals 32 meters, M P equals 81.9 meters, and M B equals x.
Answers:
(a) x = 94 centimeters
(b) x = 42.9 meters
Explanations:
(a) We'll use the angle bisector theorem in this part which states that
[tex] \frac{\text{length of }YK}{\text{length of }YV} = \frac{\text{length of }TK}
{\text{length of }VT} [/tex] (1)
Note that we put the length of YK in the numerator in equation (1) because
x = (length of VK) = (length of YK) + (length of YV) (2)
To solve for the length of YK, we multiply both sides of equation (1) by the denominator of the left side of equation (1) and so
[tex] YK = \frac{(\text{length of }TK)(\text{length of }YV)}
{\text{length of }VT} [/tex]
Since the length of TK = 168 cm, the length of YV = 34 cm, and the length of VT = 95.2 cm, the length of YK is given by
[tex]\text{length of }YK = \frac{(\text{length of }TK)(\text{length of }YV)} {\text{length of }VT} = \frac{(168)(34)} {(95.2)}
\newline \boxed{\text{length of }YK = 60}[/tex]
Recall that the length of YV = 34 cm. So, using equation (2):
x = (length of VK) = (length of YK) + (length of YV)
x = 60 + 34
x = 94 cm
(b) Since segment AB is parallel to segment NP and intersects the other sides of triangle MNP (note that A is in segment MN and B is in segment MP), using the triangle proportionality theorem,
[tex] \frac{MB}{BP} = \frac{MA}{AN} [/tex] (1)
Recall that MB = x and AN = 32. To obtain BP and MA, note that
MN = MA + AN = 67.2 (2)
MP = MB + BP = x (3)
Subtracting all sides of equation (2) by MA, we get
AN = MN - MA = 67.2 - 32
AN = 35.2
Likewise, we subtract all sides of equation (3) by MB so that
BP = MP - MB
BP = 81.9 - x
Now, we substitute the values of MA, MB, MP and AN to equation (1) and we have
[tex] \frac{x}{81.9 - x} = \frac{35.2}{32} = 1.1
[/tex] (4)
To remove fractions, we multiply all sides of equation (4) so that
x = 1.1(81.9 - x)
x = 90.09 - 1.1x
x + 1.1x = 90.09
2.1x = 90.09
x = 42.9 cm
