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Three Sample SAT Questions


Date: 11/12/96 at 19:00:38
From: mike
Subject: SAT math

Please help me with the following SAT questions: 

(1)  How do you find the diagonal in a square?

(2)  If (x+2)^2 = 25 and x>0 what is the value of x^2?

(3)  How do I do this one: 4 over n, 5 over n, and 7 over n.  It asks 
if each of the fractions above is in its simplest reduced form, which 
of the following could the value be for n? 


Date: 12/12/96 at 19:00:18
From: Doctor Daniel
Subject: Re: SAT math

Hi Mike, 

I'll answer each of your questions in order.

(1)  To find the diagonal in a square, the short answer is the 
Pythagorean theorem.  The long answer is:  

First, what's the diagonal of a square? It's the length of the line 
segment from one corner of the square to the opposite corner. The 
angles of a square are all right angles, so this would be the length 
of the hypotenuse of a right triangle whose other two sides are the 
same length as each other (which is the length of a side of the 
square).

Suppose the length of the side of the square is x and the length of 
the diagonal is d. Then by the Pythagorean Theorem, we have that:

x^2 + x^2 = d^2
2x^2 = d^2

Taking the positive square root of both sides:
 
d = sqrt (2) * x.  

So the diagonal is equal to the square root of two times the length of 
a side of the square.

(2)  If we want to find the value of x^2 in this equation, suppose we 
take the square root of both sides. Then we have:
 
x+2 = plus or minus sqrt (25).  

Suppose x+2 = + sqrt (25).  Then x+2 = 5, and x = 3. 
Suppose x+2 = - sqrt (25).  Then x+2 = -5, and x = -7.  
But they told us that x > 0.  So x must equal 3, so x^2 = 9.  

Also, since you this is from the SAT, another trick would be to plug 
the positive square roots of all the answers they give for x^2 into 
the (x+2)^2, and see if they give you 25.  This should get you the 
right answer without actually having to solve for x.

(3) Since you didn't give the listed choices for answers, I can't 
totally answer this question. But it should be clear that, if 4/n is 
fully reduced, then 2 is not a factor of n. Suppose 2 was a factor of 
n; say n was 18. Then 4/18 wouldn't be reduced; we could divide top 
and bottom both by 2. Similarly, 5 can't be a factor of n, and 
neither can 7. So suppose the possible answers were:

A) 18, B) 21, C) 35, D) 11, E) 16

Then the correct answer is D); 2 is a factor of A and E so they won't 
work and 7 is a factor of B and C, so they won't work either. 11 is 
prime, so it certainly doesn't have 2, 5, or 7 as a factor, which 
means that it is the correct answer.

I hope this helps.  Good luck on the test!

-Doctor Daniel,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Basic Algebra

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