Bosco

In rhombus ABCD, AC = 14 and AB = 20. Find the perimeter of ABCD.    

The perimeter of a rhombus in terms of diagonals "p" and "q" is P = 2 • sqrt(p² + q²)    

P = 2 • sqrt(14² + 20²)    

P = 2 • sqrt(196 + 400)    

P = 2 • sqrt(596) = 48.826 = 48.8    

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The area of a trapezoid is $80$. The length of one base is $16$ units greater than the other base, and one base of the trapezoid is $12$. Find the length of the median of the trapezoid.    

M = [ (12) + (12 + 16) ] / 2    

M = 40/2    

M = 20    

The median of a trapezoid is the average of its two bases.    

In this particular problem, the trapezoid's area is irrelevant.    

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A line and a circle intersect at A and B as shown below. Find the distance between and A and B.    
The circle is x^2 + y^2 = 1, and the line is y = x.    

x² + y² = 1 draws a circle with radius 1, centered at (0,0).    

y = x draws a straight line through (0,0).    

Since the line goes through the center of the circle, it creates a diameter.    

Since the circle has radius 1, its diameter, the distance between A & B = 2.    

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The area of a trapezoid is $80$. The length of one base is $16$ units greater than the other base, and one base of the trapezoid is $12$. Find the length of the median of the trapezoid.    

M = [ (12) + (12 + 16) ] / 2    

M = 40/2    

M = 20     

The median of a trapezoid is the average of its two bases.    

In this problem, the area of the trapezoid was irrelevant.    

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Find a six-digit multiple of $64$ that consists only of the digits $2$ and $4$.    

64 • 3,816 = 244,224     

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There are four cities: Capital City, Little Village, Mytown, and Yourtown.  The distance from Capital City to Little Village is $5$ miles.  The distance from Capital City to Mytown is $2$ miles.  The distance from Mytown to Yourtown is $3$ miles.  The distance from Little Village to Yourtown is $4$ miles.  Find the distance from Capital City to Yourtown, in miles.    

Little Village ----- 4 ----- Your Town ----- 1 ----- Capital City ----- 2 ----- My Town    

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One root of the quadratic equation $x^2 + bx -28 = 4x + 14$ is $-7$. What is the value of $b$?    

x² + bx – 28 = 4x + 14    

Change that "bx" to "kx" because I have another use for "b".    

Arrange terms in standard quadratic format ax² + bx + c = 0     

x² + kx – 4x – 42 = 0     

x² + (kx – 4x) – 42 = 0   

x² + (k – 4) x – 42 = 0      and it's given that one root is – 7    

Since one root is – 7, the other root has to be + 6 to get that – 42     

so, factored, we have (x + 7)(x – 6) which multiplied out is x² + x – 42            

& so, (k – 4) x = x which means that k = + 5 or, reverting to the words of the problem, b = 5    

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