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In the diagram, the area of the large square is 1 square unit. Two diagonal segments divide the square into four equal-sized traingles. Two of these triangles are divided into smaller red and blue traingles that all have the same height and base length. Find the area of a red traingle

User Eppye
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Answer:

Hence, the area of each small red triangle is 1/8 square units.

Explanation:

Let's denote the base length and height of each small red and blue triangle as 'b' and 'h', respectively.

Since the large square has an area of 1 square unit, each side of the square has a length of 1. Therefore, the diagonal of the square (represented by the dotted line in the diagram) has a length of √2.

The diagonal segments divide the square into four equal-sized triangles, so each of these triangles has an area of (1/4) square units.

Let's focus on one of the triangles, which consists of a red and a blue triangle.

The total area of the triangle is (1/4) square units.

The area of the blue triangle can be calculated using the formula for the area of a triangle: (1/2) * base * height.

The base of the blue triangle is the side length of the large square, which is 1 unit.

Let's denote the height of the blue triangle as 'h_blue'.

Therefore, the area of the blue triangle is (1/2) * 1 * h_blue = (1/2) * h_blue square units.

Since the red and blue triangles have the same base length and height, the area of the red triangle is also (1/2) * h_blue square units.

Now, let's consider the larger triangle (consisting of a red and a blue triangle) again. The total area of this triangle is (1/4) square units.

We know that the area of the blue triangle is (1/2) * h_blue square units. Similarly, the area of the red triangle is also (1/2) * h_blue square units.

Therefore, the sum of the areas of the red and blue triangles in the larger triangle is (1/2) * h_blue + (1/2) * h_blue = h_blue square units.

Since the total area of the larger triangle is (1/4) square units, we have the equation:

h_blue = (1/4)

Therefore, the height of the blue triangle is (1/4) unit.

As the height of the red triangle is also equal to the height of the blue triangle, the area of the red triangle is:

Area of the red triangle = (1/2) * h_red = (1/2) * (1/4) = 1/8 square units.

User Rian Mostert
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