1. Nine points are arranged in a 3 x 3 grid, as shown below. In how many different ways can you choose three points, so that they form a triangle?
2. A right isosceles triangle has a hypotenuse of 1. What is the area of the triangle?
+13 1. There are 9 dots, and to make triangles, you need 3.
C(9, 3) is 84. However, out of these 84 ways, there are 3 horizontal lines, 3 vertical, and two diagonal (these aren't considered triangles).
Now, we can use complementary counting to find the remaining number of true cases.
84 - (3 + 3 + 2) = 84 - 8 = 76.
Answer: 76 triangles
2. If a right isosceles triangle were to have a hypotenuse of 1, we could find the values of the base and height using the Pythagorean Theorem.
\(x^2 + y^2 = 1^2 = 1\), \(x^2 + y^2 = 1\).
Now because the sides are the same length, we can simplify this equation to
\(2x^2 = 1\)
\(x^2 = \frac{1}{2}\), \(x = \sqrt{\frac{1}{2}}\).
To find the area, we use the area equation of a triangle (\(\frac{bh}{2}\)). Because the base and height are the same, the equation becomes \(\frac{b^2}{2}\) or \(\frac{h^2}{2}\).
Plugging the value we got into the equation, we get
\(\frac{\sqrt{\frac{1}{2}}^2}{2} = \frac{\frac{1}{2}}{2} = \frac{1}{4}.\)
Answer: \(\mathbf{\frac{1}{4}}\)
