4.5 Answer ExplanationsGeometryDPS01120BAc
＼炉FAs shown in the figure above, if B is on the line
that bisects L.ACD then the degree measure of
90
乙DCBis—= 45. Then because B, C, and F
2
are collinear, the sum of the degree measures
of乙BCD and乙DCFis 180. Therefore,
x = 180 - 45 = 135 and 1 30::; 135 < 140.1h e correct answer 1s E.
PS07380- A rope 20.6 meters long is cut into two pieces. If the
length of one piece of rope is 2.8 meters shorter than
the (^) length of the other, what is the length, in (^) meters, of
the longer piece of rope?
(A) 7.5
(B) 8. 9
(C) 9. 9
(D) 10.3
(El 11.^7
Algebra
If x represents the length of the longer piece of
rope, 如n x - 2.8 represents the length of the
shorter piece, where both lengths are in meters. The
total length of the two pieces of rope is 20.6 meters
so,
x + (x - 2.8) = 20.6 given
2x - 2.8 = 20.6 add like terms
2x = 23.4 add 2.8 to both sides
x = 11. 7 divide both sides by 2
Thus, the length of the longer piece of rope is
1 1. 7 meters.
The correct answer is E.
36. If x and y are (^) integers and x -y is odd, which of the
following must be true?
I. xy is even.
II. x^2 + y^2 1s odd.
Ill. (x + y)^2 is even.
(A) I only
(Bl II only
(Cl Ill only
(D) I and II only
(El I, II, and Ill
Arithmetic
We are given that x and y are integers and
that x - y is odd, and then asked, for various
operations on x and y, whether the results of the
operations are odd or even. It is therefore useful to
determine, given that x - y is odd, whether x and
y are odd or even. If both x
and y are even一that is, divisible by 2 —then
x - y = 2m - 2 n = 2 (m - n) for integers m and^ n.
We thus see if both x and y are even then x - y
cannot be odd. And because x - y is odd, we see
that x and y cannot both be even. Similarly, if
both x and y are odd, then, for integers j and k, x =
2j + 1 and y = 2 k + !. Therefore, x - y = (2j
- 1 ) - (2k + 1). The ones cancel, and we are left
with x - y = 2j - 2 k = 2 (j - k). Because 2 (j - k)
would be even, x and y cannot both be odd if x - y
is odd. It follows from all of this that one of x or y
must be even and the other odd.
Now consider the statements I through III.
I. If one of x or y is even, then one of x or y is
divisible by 2. It follows that xy is divisible
by 2 and that xy is even.
II. Given that a number x or y is odd—not
divisible by 2 —we know that its product
with itself is not divisible by 2 and is
therefore odd. On the other hand, given
that a number x or y is even, we know that
its product with itself is divisible by 2 and is
therefore even. The sum x2 + y2 is therefore
the sum of an even number and an odd
number. In such a case, the sum can be