Answer:
5. x = 7
6. x = 13; YZ = 28
Explanation:
It can help to start by writing the similarity statement for each figure. To do that, you can identify corresponding features, then work around the figure in a direction that maintains the correspondence of the vertices. Here the figures are dilated, and problem 5 has the figure rotated. Since there is no reflection involved, the corresponding vertices remain in clockwise order.
Once you have written the similarity statement, you can identify the values given for corresponding parts. Here, the values are all for side lengths (not angles). Corresponding sides will have their vertices in the same relative position in the similarity statement.
5.
Working counterclockwise from the acute angle, we have ...
KNML ~ PSRQ
Then corresponding sides are ...
KN = 7x -9 . . . . PS = 60
NM = not given . . . . SR = not given
ML = 4x +4 . . . . RQ = 48
LK = not given . . . . QP = not given
Now, you can write a proportion that involves corresponding sides. The main point here is to put corresponding parts in the same relative locations in the proportion. We can write this a couple of ways:
KN/ML = PS/RQ ⇔ (7x -9)/(4x +4) = 60/48
or
KN/PS = ML/RQ ⇔ (7x -9)/60 = (4x +4)/48
We note here that solution is generally easier if the variable is put in the numerator of the expression. We might choose the latter form for that reason. We might choose the first form because we know we can reduce the fraction 60/48 to 5/4, simplifying the problem a bit.
Using the second form of the proportion, we can solve it as follows:
(7x -9)/60 = (4x +4)/48
4(7x -9) = 5(4x +4) . . . . . . multiply by the LCM of 60 and 48, which is 240
28x -36 = 20x +20 . . . . . eliminate parentheses
8x = 56 . . . . . . . . . . . add 36-20x to both sides
x = 7 . . . . . . . . . divide by 8
__
6.
Counterclockwise from S, the similarity statement can be written ...
ΔSYZ ~ ΔSRT
Corresponding parts are ...
SY = not given . . . . SR = not given
YZ = 2x +2 . . . . RT = 3x -7
ZS = 40-5 = 35 . . . . TS = 40
A suitable proportion is ...
(2x +2)/35 = (3x -7)/40
8(2x +2) = 7(3x -7) . . . . . . . . multiply by 280
16x +16 = 21x -49 . . . . . . eliminate parentheses
65 = 5x . . . . . . . . . add 49-16x to both sides
x = 13 . . . . . . . divide by 5
YZ = 2x +2 = 2(13) +2
YZ = 28