Question

Assessment items 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. What is the value of x? Enter your answer, as a decimal, in the box. m 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.

Answer 1

The answer to this question should be 39.


Hope this help you guys

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Answer 2

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 

$\frac{\text{length of }YK}{\text{length of }YV} = \frac{\text{length of }TK} {\text{length of }VT}$  (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

$YK = \frac{(\text{length of }TK)(\text{length of }YV)} {\text{length of }VT}$

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

$\text{length of }YK = \frac{(\text{length of }TK)(\text{length of }YV)} {\text{length of }VT} = \frac{(168)(34)} {(95.2)} \newline \boxed{\text{length of }YK = 60}$

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, 

$\frac{MB}{BP} = \frac{MA}{AN}$  (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 

$\frac{x}{81.9 - x} = \frac{35.2}{32} = 1.1$  (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