What is the length of line segment $AB$ in the figure

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AAPL: Moderator; Posts: 2632; Joined: Sun Oct 29, 2017 2:08 pm; Followed by: 2 members

What is the length of line segment $AB$ in the figure

by AAPL » Mon Jul 08, 2019 2:05 pm

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Economist GMAT

What is the length of line segment $AB$ in the figure above?

A. $\frac{\sqrt{2}}{2}$

B. $\frac{\sqrt{3}}{2}$

C. $\sqrt{2}$

D. $\frac{3}{2}$

E. $\sqrt{3}$

OA C

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Source: Beat The GMAT — Problem Solving | Mon Jul 08, 2019 2:05 pm

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 am; Location: Montreal; Thanked: 1090 times; Followed by: 355 members; GMAT Score:780

by Ian Stewart » Tue Jul 09, 2019 7:02 am

If we draw a vertical height from A down to BC, that divides the triangle into a 45-45-90 triangle on the left, and a 30-60-90 triangle on the right. The hypotenuse of the 45-45-90 is AC, so is of length 1, and the height we've drawn is thus of length 1/âˆš2 = âˆš2/2 (since the sides of a 45-45-90 are in a 1 to 1 to âˆš2 ratio). This is opposite the 30 degree angle in the 30-60-90 triangle, so it is the shortest side of that triangle, and the longest side AB is twice as long (since in a 30-60-90 triangle, sides are in a 1 to âˆš3 to 2 ratio), so the length of AB is 2(âˆš2/2) = âˆš2.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by Scott@TargetTestPrep » Thu Jul 11, 2019 6:51 pm

AAPL wrote:Economist GMAT

What is the length of line segment $AB$ in the figure above?

A. $\frac{\sqrt{2}}{2}$

B. $\frac{\sqrt{3}}{2}$

C. $\sqrt{2}$

D. $\frac{3}{2}$

E. $\sqrt{3}$

OA C

If we drop a perpendicular from vertex A to side BC, that is, if we draw the height, we will divide the triangle into two special right triangles: a 45-45-90 triangle on the left and a 30-60-90 triangle on the right. Let's call this height AD; that is, D is a point on BC such that AD is perpendicular to BC.

We know that the side ratio of a 45-45-90 triangle is x : x : xâˆš2. Since AC = 1, we see that if we let AD = x, then xâˆš2 = 1. So x = 1/âˆš2 = âˆš2/2 = AD.

We also know that the side ratio of a 30-60-90 triangle is x : xâˆš3 : 2x. Since AD = âˆš2/2, we see that AB must be twice as much. So AB = 2(âˆš2/2) = âˆš2.

Answer: C

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