From exterior point L of circle O, tangent segments LP and LQ are drawn such that the measure of angle PLQ is 60 degrees. If a r
adius of circle O measures 6, what is the distance from the center of the circle to point L?
1 answer:
Refer to the attached diagram for help.
All radii of a circle are congruent.
Tangent segs LP and LQ are congruent. (Tangent segs drawn to a circle from the same point are congruent)
Because of this we can establish OPLQ as a kite.
OPLQ is a kite implies that OL bis. angle PLQ.
An angle bis. divides an angle into 2 congruent, parts, so angle OLP must be 30 degrees.
Since tangents form right angles with the radius, angle OPL is right.
Now we have a right triangle OPL with a 30-degree angle. The other angle must be 60 degrees because of the no-choice theorem. (180-(30+90)) = 60
We know that OP is 6 because it is a radius of circle O. How do we use that to find OL? Well, we <em>could</em> use trig, but you might recognize this as a special triangle!
The side opposite the 30 degree angle = x
The side opposite the right angle = 2x
So OL must be 12!
All radii of a circle are congruent.
Tangent segs LP and LQ are congruent. (Tangent segs drawn to a circle from the same point are congruent)
Because of this we can establish OPLQ as a kite.
OPLQ is a kite implies that OL bis. angle PLQ.
An angle bis. divides an angle into 2 congruent, parts, so angle OLP must be 30 degrees.
Since tangents form right angles with the radius, angle OPL is right.
Now we have a right triangle OPL with a 30-degree angle. The other angle must be 60 degrees because of the no-choice theorem. (180-(30+90)) = 60
We know that OP is 6 because it is a radius of circle O. How do we use that to find OL? Well, we <em>could</em> use trig, but you might recognize this as a special triangle!
The side opposite the 30 degree angle = x
The side opposite the right angle = 2x
So OL must be 12!
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Answer: 12 dimes and 5 quarters
Step-by-step explanation: Let's start this problem by setting up a chart so we can organize our information that we're given in this problem.
Down the left side, we'll have our different types of coins. In this case dimes and quarters. Across the top we'll have our formula which is shown below.
Number of coins · value of each coin = total value
Now let's fill out our chart.
For number of dimes and quarters, we know that mark has a total of 17 dimes and quarters but we don't know how many of each he has. In fact, that's what the problem is asking.
So if we represent our number of dimes as <em>x</em>, we can call our number of quarters <em>17 - x</em>.
The value of each dime is 10¢ and the value of each quarter is 25¢.
Our total value based on our formula is going to come from the first column times the second column. So the total value of our dimes is <em>x times 10</em> or <em>10x </em>and the total value of our quarters is <em>17 - x times 25</em> or <em>25 (17 - x)</em>.
Our goal in this problem will be to find <em>x </em>because <em>x</em> represents our number of dimes and the problem asks how many dimes does he have. If we know the number of dimes, we can easily find the number of dimes and we'll have our answer. However, we need an equation in order to find <em>x</em>.
It's important to understand that he information in this equation will always come from the last column of your chart, the total value column. So what do we know about the total value of our dimes and the total value of our quarters?
Well we know that the total value of all of our coins is $2.45. So if we add the total value of our dimes + the total value of our quarters, that should equal $2.45. So below the chart I added an additional box and I put 245 in it. Notice that I wrote 245 in terms of cents because our value of dimes and value of quarters is also written in terms of cents and we need to be constient.
So here's our equation.
10x + 25 (17 - x) = 245.
If we solve this, we get <em>x = 12</em>.
Going back up into our chart, remember that <em>x</em> represents our number of dimes so Mark has 12 dimes. To get his number of quarters, we take 17 - x which is 17 - 12 or 5 quarters and that's our answer.
I have also attached the chart that I have made below.
