2.
For x > 0, the perimeter of the triangle with vertices (0, 0), (x, 0) and (x, 5)
is 30 units. What is the value of x?
2.
We don’t know the value of x, but we can still graph this
triangle; we just don’t know how far the triangle actually
stretches from left to right. From the coordinates we do know
it is a right triangle. The horizontal side is x units and the
vertical side is 5 units. Because this is a right triangle, we can
determine the length of the hypotenuse, c, in terms of x, with
the help of the Pythagorean Theorem:
=
+
so c =
. The perimeter is 30 units, therefore:
x+5+
= 30;
= 25 – x;
- 25 = 625 – 50x +
; 50x = 600;
x = 12. Notice this triangle has the rare quality that the value of its perimeter is equal to
the value of its area.
4.
The area of a rectangular patio is 360 square meters. Surrounding the patio
on all four sides is a flower border 2 meters wide as shown in the diagram.
The border has an area of 184 square meters. What is the number of
meters
in
the
perimeter of
the patio?
4.
At every corner of the patio, there is a 2 meter by 2 meter square portion of flower bed
that does not directly abut the perimeter of the patio. This amounts to 4 square meters of
flower bed at each of four corners for a total of 16 square meters. If we subtract 16 square
meters from the 184 square meters of flower bed, we have 168 square meters of flower bed
remaining. This is exactly double the perimeter of the patio, since the border is 2 meters
wide. The perimeter of the patio is thus 168 2 = 84 meters.
10. Challenge. You form a rhombus by putting two equilateral triangles with
side length 2n together, as shown. Write an expression for the area of the
rhombus in terms of n. Explain your reasoning.