MA.8.G.2.1

Use similar triangles to solve problems that include height and distances

Block 30

Pre-test

Work on the problems on the worksheet “Indirect Measurement” Why do you think it is called indirect measurement?

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Indirect measurement

Methods for obtaining measurements of properties of figures, particularly distances, which are inaccessible to direct measurement.

• The work of engineers, surveyors, cartographers, and astronomers would be rendered virtually impossible without indirect measurement schemes. • The branch of mathematics known as trigonometry arose as a response by ancient Greek mathematician-astronomers to the problem of obtaining inaccessible measurements: the distances and angles between certain celestial bodies.

Eratosthenes of Cyrene (276 -194 BC)

Contributions: • The Sieve of Eratosthenes as a way of finding prime numbers. • The measurement of the Sun-Earth distance, now called the astronomical unit (804,000,000 stadia, 1 stadion varies from 157 to 209 meter). • The measurement of the distance to the Moon (780,000 stadia). • The measurement of the circumference of the Earth

Earth’s Circumference

• • He used trigonometry and knowledge of the angle of elevation of the Sun at noon in Alexandria and Syene (now Aswan, Egypt). The calculation is based on the assumption that the Earth is spherical and that the Sun is so far away that its rays can be taken as parallel.

• • • • • • At local noon the Sun would appear directly overhead in Syene Using a vertical stick and measuring the cast shadow, in his hometown of Alexandria, the angle of elevation of the Sun would be 83° at the same time. Using geometry of parallel lines, the distance from Alexandria to Syene must be 7/360 of the total circumference of the Earth.

The distance between the cities was known to be about 5,000 stadia. He established a final value of 700 stadia per degree, which implies a circumference of 252,000 stadia.

Eratosthenes' value corresponds to between 39,690 km and 46,620 km. The circumference of the Earth around the poles is now measured at around 40,008 km.

Eratosthenes' Experiment

Using Shadows

A person 5 feet 3 inches tall casts a 6-foot shadow. At the same time of day, a lamppost casts an 18-foot shadow. What is the length of the lamppost?

Explain how this method works

Assumptions in this method

• • • • • Objects are standing straight or leaning at the same angle Land upon which both shadows fall is level Sun strikes the top of the objects at the same angle Distance source of light, light rays are parallel to each other Closer source of light, the terminal ends of the shadows must be equidistant from the light source

No shadows available

Using mirrors instead. Why does this method work?

What information is necessary for this type of problems?

Design a method

• A guy wire attached to the high tower needs to be replaced. The contractor does not know the height of the tower or the length of the wire. Find a method to measure the length of the wire indirectly

Distance across the river

Explain this method

Kristin method uses string and a ruler. She tacks a piece of string to the base of the wall and walks back from the wall holding the other end of the string to her eye with her right hand.

She holds a 12-inch ruler parallel to the wall in her left hand and adjusts her distance to the wall until the bottom of the ruler is in line with the bottom edge of the wall and the top of the ruler is in line with the top edge of the wall. Now with two measurements, she is able to calculate the height of the room. Explain her method. If the distance from her eye to the bottom of the ruler is 23 inches and the distance from her eye to the bottom of the wall is 276 inches, calculate the height of the room.

New Method

The Thumb method

• • Indirectly determines the distance to an object Uses also similar triangles

Distance to a car whose side is facing you

1. Face the car directly. Do not stand sideways or diagonally 2. Straighten your right arm in front of you and perpendicular to your body and extend your right thumb upward 3. Close your left eye, and align your thumb with the left end of the car. Your right eye to your thumb to the left end of car will form a straight line

4. Without moving – keep your body still and do not move your head or arm – switch eyes (close the right eye, open the left eye). Notice how your thumb now marks a different spot on the car. It appears to have moved to the right, and not be in the car at all. Estimate the distance your thumb appears to move.

5. Multiply the distance by 10. That is the distance to the car.

Explain the Geometry behind the thumb method Why do we multiply by a factor of 10 in the last step?

Trying it out

• • • • Go to the parking lot Determine what car is 300 feet or 500 feet away Cars are typically 15 feet long Try the method yourself

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Questions

What happens if you are not facing the object straight on? Will the method still work?

What happens if you do not keep your head still while switching eyes?

What is the object is oriented diagonally? What would you do differently if you know the distance to the object but not its size?

How would you use this method to measure height?

Double Indirect Measurement

Suppose you want to know the width of a large river, but there are no objects of known size on the opposite bank to help estimate how far the thumb moves

1. Use the thumb method and remember the two points marked by your thumb (R and S). The distance between those two points is b.

2. Mark the point where you are standing P 3. Walk backwards along the line created by your right eye and thumb 4. As you walk backwards, the distance between the two points marked by your thumb increases. Stop when this distance is twice b.

5. Mark your new location, point Q 6. The distance from point P to point Q is the same as a, the distance we were looking for.

Geometric explanation of Double Indirect Measurement

Can you explain why it works?

GeoGebra file: double_indirect

• • • Move point Q to determine the effectiveness of the double indirect method.

Move points R and S to change the original triangle Is it necessary for the distance k to be the same as b? Could k be a fraction of b?