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# Counting

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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?

Oct 24, 2021

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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.

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: $$\mathbf{\frac{1}{4}}$$

Oct 24, 2021