Triangles ABC and DEC are similar. Side DE is 1.8 feet, side EC is 4 feet, and side BE is 0.8 foot.

Part A: Describe how to draw triangles ABC and DEC separately. Be sure to give the lengths of sides BC, DE, and EC in the description.

Part B: Find side AB. Show each step of your work.

The lengths of sides BC, DE, and EC in triangle DEC are given as [tex]4[/tex] feet, [tex]1.8 feet,[/tex] and 0.8 foot, respectively. The length of side [tex]AB[/tex] is approximately [tex]3.36[/tex] feet.

Part A:

To draw triangles  [tex]ABC[/tex] and  [tex]DEC[/tex] separately, follow the steps below:

Draw a straight line segment DE with length  [tex]1.8[/tex] feet.

From point D, draw a line segment DC perpendicular to DE with length 4 feet.

Connect point E and point C with a straight line segment EC.

From point B, draw a line segment BE perpendicular to BC with length 0.8 foot.

Draw a straight line segment BC that intersects line segment DC at point A.

Connect point A and point B with a straight line segment AB.

The lengths of sides BC, DE, and EC in triangle DEC are given as [tex]4[/tex] feet, [tex]1.8 feet,[/tex] and 0.8 foot, respectively.

Part B:

Since triangles  [tex]ABC[/tex] and [tex]DEC[/tex] are similar, their corresponding sides are proportional. Let x be the length of side AB. Then we have:

[tex]AB/DE = BC/EC[/tex]

Substituting the given values, we get:

[tex]x/1.8 = BC/0.8[/tex]

Solving for BC, we get:

[tex]BC = 0.8x/1.8[/tex]

Since BC and EC are the two legs of a right triangle DEC, we can use the Pythagorean theorem to find BC:

[tex]BC^2 + EC^2 = DE^2[/tex]

Substituting the given values, we get:

[tex](0.8x/1.8)^2 + 4^2 = 1.8^2[/tex]

Solving for x, we get:

[tex]x = 1.8 * (sqrt(1.8^2 - 4^2)) / 0.8[/tex]

[tex]x \approx 3.36 feet[/tex]

Therefore, the length of side [tex]AB[/tex] is approximately [tex]3.36[/tex]  feet.

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