Rectangle ABCDABCDA, B, C, D is graphed in the coordinate plane. The following are the vertices of the rectangle: A(5, 5),A(5,5),A, left parenthesis, 5, comma, 5, right parenthesis, comma B(7,5)B(7,5)B, left parenthesis, 7, comma, 5, right parenthesis, C(7,-1)C(7,−1)C, left parenthesis, 7, comma, minus, 1, right parenthesis, and D(5,-1)D(5,−1)D, left parenthesis, 5, comma, minus, 1, right parenthesis. Given these coordinates, what is the length of side BCBCB, C of this rectangle?

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do you have any choices

step-by-step explanation:

∞ this is your sign

, and

for

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

note the following:

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

, and for

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4