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2 years ago

7

Triangle AOC intersects a circle with center O. side AO is 10 inches and the diameter of the circle is 12 inches. What is the le

ngth of BC

1 answer:

zvonat [6]2 years ago

4 0

Answer:

$AC=8inches$

Step-by-step explanation:

It is given that triangle AOC intersects a circle with center O, side AO is 10 inches and the diameter of the circle is 12 inches, thus

OC is the radius of the circle and is equal to $\frac{D}{2}=\frac{12}{2}=6 inches$ .

Now, From ΔAOC, using the Pythagoras theorem, we get

$(OA)^{2}=(OC)^{2}+(AC)^{2}$

⇒ $(10)^{2}=(6)^2+(AC)^2$

⇒ $100=36+(AC)^2$

⇒ $64=(AC)^2$

⇒ $AC=8inches$

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3 years ago

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$t_{10} = -22 \ \text{ and } S_{10} = -85$

========================================================

Explanation:

$t_1 = \text{first term} = 5\\t_2 = \text{first term}-3 = t_1 - 3 = 5-3 = 2$

Note we subtract 3 off the previous term (t1) to get the next term (t2). Each new successive term is found this way

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We plug in $t_1 = 5 \text{ and } d = -3$ and simplify

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Then we can plug in various positive whole numbers for n to find the corresponding $t_n$ value. For example, plug in n = 2

$t_n = -3n+8\\t_2 = -3*2+8\\t_2 = -6+8\\t_2 = 2$

which matches with the second term we found earlier. And,

$tn = -3n+8\\t_{10} = -3*10+8\\t_{10} = -30+8\\t_{10} = \boldsymbol{-22} \ \textbf{ is the tenth term}$

---------------------

The notation $S_{10}$ refers to the sum of the first ten terms $t_1, t_2, \ldots, t_9, t_{10}$

We could use either the long way or the shortcut above to find all $t_1$ through $t_{10}$ . Then add those values up. Or we can take this shortcut below.

$Sn = \text{sum of the first n terms of an arithmetic sequence}\\S_n = (n/2)*(t_1+t_n)\\S_{10} = (10/2)*(t_1+t_{10})\\S_{10} = (10/2)*(5-22)\\S_{10} = 5*(-17)\\\boldsymbol{S_{10} = -85}$

The sum of the first ten terms is -85

-----------------------

As a check for $S_{10}$ , here are the first ten terms:

t1 = 5
t2 = 2
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t9 = -19
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Then adding said terms gets us...

5 + 2 + (-1) + (-4) + (-7) + (-10) + (-13) + (-16) + (-19) + (-22) = -85

This confirms that $S_{10} = -85$ is correct.

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