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In the figure above, line segment CZ = 6. What is the

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Source: — Data Sufficiency |

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by Jay@ManhattanReview » Wed Apr 10, 2019 11:50 pm

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VJesus12 wrote:Image

In the figure above, line segment CZ = 6. What is the length of line segment AB?

(1) Line segment XY = 4
(2) Line segment AZ > Line segment AY

[spoiler]OA=A[/spoiler]

Source: Veritas Prep
Let's take each statement one by one.

(1) Line segment XY = 4

You may observe that ∆XYZ ≈ ∆BAZ; thus,

YX/YZ = BA/AZ => YX/YZ = BA/AZ => 4*AZ = BA*YZ---(1)

You may also observe that ∆CZA ≈ ∆BAY; thus,

CZ/YZ = BA/YA => CZ/YZ = BA/YA => 6*YA = BA*YZ ---(2)

From (1) and (2), we have 4*AZ = 6*YA => AZ = 1.5*YA => YZ = AZ + YA = 1.5YA + YA => YZ = 2.5YA

Again, from (1): 4*AZ = BA*YZ, we have 4*1.5*YA = BA*2.5*YA => BA = 2.4. Sufficient.

(2) Line segment AZ > Line segment AY

Clearly insufficient.

The correct answer: A

Hope this helps!

-Jay
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by deloitte247 » Fri Apr 26, 2019 12:17 am

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What is the Length of Line segment AB ?
Triangle XZY is similar to Triangle BZA and Triangle CYZ is similar to Triangle BYA
$$Ratio=\frac{XZ}{BZ}=\frac{ZY}{ZA}=\frac{XY}{AB}=\frac{CY}{BY}=\frac{YZ}{YA}=\frac{CZ}{BA}$$

Statement 1
Line segment XY=4, Using
$$\frac{ZY}{ZA}=\frac{XY}{AB}\ and\ \ \frac{YZ}{YA}=\frac{CZ}{AB}$$
$$ZY=YA+YZ\ \ XY=4\ CZ=6$$
$$\frac{YA+AZ}{AZ}=\frac{4}{AB}------------eqn\left(i\right)$$
$$\frac{YA+AZ}{YA}=\frac{6}{AB}------------eqn\left(ii\right)$$
eqn i divided by eqn ii
$$\frac{YA+AZ}{AZ}\cdot\frac{AZ\ AY}{\frac{YA}{AZ}}=\frac{4}{AB}\cdot\frac{AB}{6}$$
$$\frac{AY}{AZ}=\frac{4}{6}=\frac{2}{3}$$
$$\frac{YA+AZ}{AZ}=\frac{4}{AB}$$
$$\frac{AY}{AZ}+\frac{AZ}{AZ}=\frac{4}{AB}$$
N.B AY/AZ = 2/3 , hence AY = 2 and AZ = 3
$$\frac{2}{3}+\frac{3}{3}=\frac{4}{AB}$$
$$\frac{5}{3}=\frac{4}{AB}$$

Statement 2
Line segment AZ > line segment AY, the exact value is not given in this statement here. Hence it is INSUFFICIENT.
$$Answer\ is\ Option\ A\ $$

Statement 1 alone is SUFFICIENT.
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