The abc toy company is creating two similar pieces for a board game, as shown below.
**find the value of x that makes the two pieces similar.**

answer:   the value of x is 14 units.

step-by-step explanation:   given that the abc toy company is creating two similar pieces for a board game as shown in the figure.  

we are to find the value of x that makes the two pieces similar.

we can see that two pieces are in the shape of triangles, δabc and δdef.

also, ab = 6 units, bc = 7 units, de = 12 units   and   ef = x = ?

for the two triangles to be similar, their corresponding sides must be proportional.

so, to make δabc and δdefδabc and δdef similar to each other, we must have

thus, the value of x is 14 units.

step-by-step explanation:

we have been given that the abc toy company is creating two similar pieces for a board game.

to find the the value of x we will use proportion as proportion states that two fractions are equivalent.

we can see that side ab corresponds to side de and side bc corresponds to side ef, so we can set a proportion for our given side lengths as:

upon substituting our given values in above equation we will get,

let us multiply both sides of our equation by 7.

therefore, the value of x equals 14 will make the two pieces similar.

, and

for

step-by-step explanation:

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

1 the 1st term, 3 the 2nd term, 6 the 3rd term,etc.

note the following:

from the 1st term (1) to the 2nd term (3), we go up by 2,from the 2nd term (3) to the 3rd term (6), we go up by 3from the 3rd term (6) to the 4th term (10), we go up by 4, andfrom the 4th term (10) to the 5th term (15), we go up by 5.

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

Xis -1/2
y is 35/4.