MATHEMATICS 157

Solutions for First Assignment

Wednesday, September 10

The three lines

form a triangle. Answer the following questions about the triangle.

1. What are the vertices?

   The vertices are .

2. What are the slopes of the sides?

   The slopes of the three sides are .

3. Which side has the steepest slope?

   Since , so the side y = 2x - 1 has the steepest slope.

4. From the slopes (only) show that the triangle is right-angled.

   Since the product of the slopes 2 and -1/2 is -1, so the angle between the two sides y = 2x - 1 and y = -1/2x + 9/2 is a right angle. So the triangle is right-angled.

5. What are the lengths of the sides?

   The lengths of the three sides are .

6. From the lengths (only) show that the triangle is right-angled.

   Since , so the trangle is right-angled.

7. What is the area of the triangle?

   The area of the triangle is 6/5.

8. How many points with integer coordinates are there inside the triangle (including on the boundary)?

   There are four points with integer coordinates inside the triangle (including on the boundary).

9. Give an equation for the circle passing through the vertices.

   The equation of the circle is .

10. Is the area of the triangle greater or lesser than of the area of the circle?

    The area of the circle is . the area of the triangle. So the area of the triangle is less than 1/4 of the area of the circle.

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