In this lesson, you've learned how to find the lengths of segments formed by chords, tangents, and secants. Let's look at a real life situation where you would need to use these new skills.
Suppose you are visiting an aquarium with a huge circular fish tank. You're standing at point C, about 6 feet away from the tank. The distance from you to another point on the tank forms a tangent that is 24 feet long. Can you use this information to figure out how wide the tank is?
Work through the steps on the slides below to solve this problem. It's a process you can use to find the width of any circular structure, such as a building, a rocket fuselage, or a labyrinth--no tape measure required!
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The first thing you should do is draw a diagram that represents your situation. Try drawing the tank and your distance from it, including the tangent.
Now, you'll need to use the rule you learned to apply when a tangent and a secant intersect. That rule will allow you to find the radius of the circle formed by the tank. What should you do with the numbers you know already to arrive at the radius? Multiply the external segment by the length of the entire segment. When there is a tangent segment, then you square that segment length. The equation you'll use is: 24² = 6(6 + 2r)
Determine the radius of the tank using the rule you recalled on the previous slide. Work out the problem in your notebook, and then check your answer below. 24² = 6(6 + 2r) The radius of the circle is 45 feet--it is 45 feet from the wall of the tank to the tank's center.
Now, think about the relationship between the radius and diameter. You can use that relationship to figure out the diameter of the tank (how wide it is). Write the width of the tank in your notebook. Then, check your answer. r = \(\mathsf{ \frac {d} {2} }\) Since the radius is 45 feet, the diameter is 90 feet. That tank is almost one hundred feet wide! That's a really large fish tank! |
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Question
You just figured out the length of a missing segment when the intersection of two segments was outside a circle. When the intersection of two segments is inside a circle, what is the rule to find a missing segment length?
If the intersection point is inside the circle, then we know that two chords have formed that intersection. The rule for using that type of intersection is: The product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.
KN(NF) = GN(NM)
