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Problem 590

# Construct the segment of length equal to $$\mathrm{a} \cdot \mathrm{b}$$ where a and $$\mathrm{b}$$ are the lengths of the segments below. (Unit length is shown below.)

Expert verified
Draw a line segment AB with lengths AC = a and BC = b. Draw circles centered at A and B with radius (a + b). Let the circles intersect at points D and E. Draw line segments DE and AD. Draw a line parallel to DE through B, intersecting circle A at F. The length of segment AF represents $$a \cdot b$$, due to similarity of triangles ADE and ABF.
See the step by step solution

## Step 1: Draw a line segment of length (a+b)

Draw a line segment AB with the lengths of segments a and b next to each other so that the length of the segment AB is equal to the sum of the lengths of segments a and b. Mark the point C where segment a ends and segment b starts, so that AC = a and BC = b. Draw a unit length segment as well, for reference.

## Step 2: Draw the circle with radius a+b centered in A and B

Using a compass, draw a circle centered at point A with radius equal to the length of segment AB (which is a + b). Also draw a circle centered at point B with the same radius.

## Step 3: Find the intersection points D and E

The circles centered at A and B will intersect at two points, D and E. Choose any one of them, for example, point D (in case the circles do not intersect, extend line segment AB until they do).

## Step 4: Draw the line segment DE

Draw the line segment connecting points D and E.

## Step 5: Draw the line segment AD

Draw the line segment connecting points A and D.

## Step 6: Draw the line parallel to DE through B

Draw a new line through point B that is parallel to line segment DE. This line will intersect the circle with center A at some point F.

## Step 7: Construct line segment AF

Draw the line segment connecting points A and F.

## Step 8: Calculation: Length of AF is equal to a · b

The length of AF represents the product a·b. This is due to the similarity of triangles ADE and ABF, which results in the following relationship: $$\frac{AF}{AD} = \frac{AB}{DE}$$. The ratio of sides DE and AB is equal to the product $$a \cdot b$$, so the length of AF = $$a \cdot b$$.

## Step 8: Complete the construction

Now that you have constructed the segment of length a · b, label this segment and provide a clear and concise explanation of the steps taken to construct this segment, as demonstrated above.

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