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# math

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Two sides of a square are divided into fourths and another side of the square is trisected, as shown. A triangle is formed by connecting three of these points, as shown. What is the ratio of the area of the shaded triangle to the area of the square? Express your answer as a fraction.

Jul 28, 2024

#2
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We can set up variables to complete this problem.

First, le'ts graph this problem. We have

Now, let's set up our first variable.

Let's let the side of square be x. Since the base of the triangle is the same length as the side of the square, we have

base of triangle = x as well.

Next, from the graph, which we nicely split into quarters, note that

height of triangle = $$\frac{3x}{4}$$

Now, since we have the base and height of the triangle, we can find it's area in terms of x. We have

$$\frac{3x}{4}\cdot x \cdot \frac{1}{2} = \frac{3x^2}{8}$$

The area of the square is just the side squared, so

Area of square = $$x^2$$

Thus, we have what we need.

$$ratio = \frac{3x^2}{8} \cdot \frac{1}{x^2} = \frac{3}{8}$$

So our answer is just 3/8.

Thanks! :)

Jul 29, 2024
edited by NotThatSmart  Jul 29, 2024
#3
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Side of square = x

Base of triangle = x

Height of triangle = 3x/4

Area of triangle = 3x^2/8

Area of square = x^2

Ratio - (3x^2)/(8x^2) = 3/8

Jul 30, 2024