MPM2D – Grade 10 Academic Math

Assignment #1: Analytic Geometry Name: _____________________________________

Teacher: _____________________________________

This assignment is due on: _____________________________________

Notes:

• Your assignment must be completed on this paper. • You must hand in your assignment on or before the due date. Once your class

is over on the day it is due your assignment is considered late, and you will be penalized. If you know you will not be in class to hand in your assignment then you must arrange to hand it in early or have someone hand it in for you. Excused absences are not acceptable reasons for late assignments.

• A completed, signed Assignment Resource Citation form (form included at the end of this assignment) must accompany your assignment. For details about what to include on this form, see your teacher.

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The Bermuda Triangle A n a l y t i c G e o m e t r y A s s i g n m e n t

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Background

The Bermuda Triangle has always been controversial. It is a region

in the Atlantic Ocean where many airplanes and ships have

mysteriously disappeared over the course of history. Some people

believe there are mysterious forces at work. Others believe that the

disappearances are coincidental, or simply a product of weather

phenomena unique to that region. This assignment will make use of

the geometry ideas covered in this course to investigate some

properties of the Bermuda Triangle.

Assignment

A researcher named Polly has been investigating the Bermuda Triangle (BT) from a small island

north of Cuba, called Skwaera which, in her charts, is situated at the origin of a Cartesian Plane.

Polly named the Bermuda Triangle as ∆ABC and determined the coordinates of the vertices:

A(1, 5) B(8,9) C(7,1)

You have come to visit Polly to learn more about the triangle, but Polly has mysteriously

disappeared. Your job is to continue her research by answering the following questions.

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1. Graph the BT on the chart below (∆ABC). Each unit on the grid represents 20 km. You will use

this diagram for the rest of the assignment.

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2. Now you must determine what type of triangle we are dealing with. (10A)

a. Is the BT a right-angled triangle? Explain your answer using mathematics.

b. Is the BT scalene, isosceles, or equilateral? Explain your answer using mathematics.

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3. In order to warn ships of potential danger, you’ve decided to put a distress beacon on a buoy

floating inside the triangle. In order to make sure that it can be seen from anywhere inside the

triangle, it must be placed equidistant from all three vertices of the triangle. (10A)

a. Geometrically speaking, what kind of point is this?

b. Your assistant has determined that the beacon should be placed at the point 11,52

.

Use analytic geometry to verify whether or not this is the correct point to place the beacon,

and if it is not, find the correct point.

Draw and label the correct point on the diagram as P.

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4. Finally! You’ve received a message that Polly is located either at 16 ,53

or 5, 92

.

You also know that one of these points is the orthocentre and the other is the centroid. But

which is which? You don’t know yet. You want to send a rescue ship to pick her up. But in order

to be able to do it, you have to find out which of these two points is the orthocentre and which

is the centroid (label the centroid as Q and the orthocentre as R on the diagram.). Determine

which point is which using algebraic techniques. (10A)

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5. If the rescue ship picks Polly up at the orthocentre and is going straight to Skwaera (which you

will recall is located at (0, 0) – the origin), determine how long the ship will be inside the BT

once Polly is on board. Assume the ship travels at a constant speed of 80 km/h. Remember that

on your graph, 1 unit = 20 km. Leave your answer in minutes, rounded to the nearest minute.(10T)

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6. This question is about all three points you found in your research of BT (P, Q, R). (5T)

a. Use analytic geometry (algebra) to show that P, Q, and R are collinear.

b. Determine the equation in standard form of the line that goes through these points.

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7. You have received an SOS signal from a ship which entered the BT going along the line

3 614

y x= + . The captain has determined that the ship is equidistant from the vertices A and C

of the BT, but due to instrument failure he cannot determine his exact location.

In order to help this ship you need to know the exact location of it. [10T]

a. On which line is the ship now. What is the equation of that line?

b. Find the coordinates of the ship algebraically.

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c. How far is the ship from you if you are on Skwaera island? Round your answer to the

nearest tenth of a kilometer.

d. On the grid below, draw the BT, the path of the ship and its location (point S).

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Assignment Resource Citation Form

This form is to be submitted with your assignment, even if you did not use any resources, or your assignment will be considered late. Your signature at the bottom confirms that all the work you submit is your own.

Remember to sign the form, and remember to have a parent/guardian sign it as well!

I did not receive or give any help for this assignment (check here)

OR

Question Number

Help I Received

- If you received help from a person, list the name of the person and the date they helped you

- If you received help from a website or book, cite the resource in MLA format

Help I Gave

- List the name of the person you helped, and the date you helped them

Your signature: _________________________________ Parent/guardian signature _________________________

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Checklist for Communication General checklist for good communication

Written Work • Solutions are easy to read • Solutions are easy to follow • Solutions are audience-appropriate • Appropriate amount of work shown • Notation is used properly • Variables defined appropriately • Conclusions written correctly • Correct precision used

Diagrams and Graphs • Straightedge used for all lines • Diagrams are sized appropriately • Diagrams are labelled properly • Graphs have the axes labelled • Graphs have the scale indicated, where appropriate • Graphs are labeled • Important points on graphs are labeled

GRADE TOTALS FOR THIS ASSIGNMENT

Application

Thinking

Communication

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