Name: ______Date: ______

Perpendicular Bisector Theorem

  1. An engineer wants to build a t-shirt stand so that it is located the same distance to the two most popular rides, the Paddle Boats and the Rollin’ Coaster.
  1. Using tracing paper, trace points P and R.
  2. Draw .
  3. Fold point P onto point R.
  4. Trace the crease onto the map above.
  5. Pick a point on the line from part d (that does not lie on ). Label this point Q.
  1. What type of line does the crease represent?
  1. Calculate PQ and RQ. How does PQ compare to RQ?
  1. Why do you get this comparison?
  1. You have to hang a board on the wall above the desk. How should the lengths of the string (from each corner of the board to the nail) compare?
  1. Why does this comparison keep the board level?

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Angle Bisector Theorem (use the diagram on the next page)

  1. A city engineer wants to build Main Street between Elm Street and Grove Street so that the shops on Main Street are the same distance to Elm Street and Grove Street.
  1. Using Tracing Paper, trace the angle formed by Elm Street and Grove Street. Label the vertex point D.
  2. Fold Elm Street onto Grove Street.
  3. Trace the crease onto the map on the next page.
  4. Pick a point on the line from part c (other than point D). Label this point F.
  5. Draw a perpendicular segment from point F to each side of the angle. Label the point on Elm Street point E and the point on Grove Street point G.
  1. What type of line does the crease represent?
  1. Calculate FE and FG. How does FE compare to FG?
  1. Why do you get this comparison?
  1. City Commissioners want to divide undeveloped land in their city into ½ residential zone and ½ business zone. The land is formed by two streets. The commissioners picked four points (points B, E, H, and K) and measured the distance those points were to each street. What point on the map below would lie on the dividing line?
  1. Why does this point lie on the dividing line?

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Name: ______Date: ______

Concurrency of Perpendicular Bisectors Theorem

  1. A businessman has three retail stores located in a city (points A, B, and C on the diagram below). The man wants to build a warehouse that will supply each of the three stores. To minimize costs and maximize effectiveness, the businessman needs to find a location that is the same distance to each of the three stores.
  1. Using a straightedge, draw ABC.
  2. Using tracing paper or constructions, draw or construct the perpendicular bisectors of two of the three sides of the triangle to the right.
  3. Label the point the bisectors intersect point D.
  4. Using a compass, construct D with a radius equal to AD.
  1. Circle the correct answers: Since point D is the center of a circle that is (circumscribed, inscribed) about a triangle, it is called the (circumcenter, incenter).
  1. How does constructing D prove that point D is the same distance from points A, B, and C?

Concurrency of Angle Bisectors Theorem

  1. A bus station is being built in an area bordered by three streets. The bus station needs to be built so that it is the same distance to each street.
  1. Using a protractor, draw the bisectors of two of the angles.
  2. Label the point the bisectors intersect point S.
  3. Using a straightedge, draw a perpendicular segment from point S to . Label the other endpoint T.
  4. Using a compass, construct S with a radius equal to ST.
  1. Circle the correct answers: Since point S is the center of a circle that is (circumscribed, inscribed) in a triangle, it is called the (circumcenter, incenter).
  1. How does constructing S prove that point S is the same distance to , , and .

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Name: ______Date: ______

Centroid Theorem

A triangular-shaped object needs to be balanced on a pole. To do this, the center of gravity of the triangle, called the centroid, must be found.

  1. In the triangle below, draw , which is one median of the triangle (a median is a segment drawn from a vertex to the midpoint of the opposite side).
  1. The Centroid Theorem states that the distance from the vertex to the centroid is 2/3rd the length of the median.
  1. Calculate AD (the length of the median).
  1. If point G is the centroid, what is AG (the length from the vertex to the centroid)?
  1. Draw point G on the triangle above.
  1. Calculate DG. How does AD compare to DG? How does AG compare to DG?
  1. Using the comparisons you made in question #2d, complete the table below: (hint: Determine the length of the shorter segment. Use that length to determine the other missing lengths)

Centroid to the Midpoint
(shorter segment) / Vertex to the Centroid
(longer segment) / Median
4 in.
14 m
42 cm
x

Triangle Midsegment Theorem

  1. Plot and label the midpoint of point S, point T, and point U.

  1. Calculate PQ and QR.
  1. Draw . Calculate SU.
  1. is parallel to which side of PQR?
  1. How does SU compare to the side of the PQR it is parallel to?

A midsegment of a triangle is a segment connecting the midpoints of two sides of a triangle. The Triangle Midsegment Theorem states that the length of a midsegment of a triangle is ½ the length of the side of the triangle the midsegment is parallel to.

  1. Use this theorem to calculate the length of TU.
  1. Calculate PR (use Pythagorean Theorem). Then calculate ST (the length of the midsegment parallel to .

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