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Geometry and Trigonometry Difficulty: Hard

The figure presents triangle A, C E, with horizontal side A, C. Side A, E, is perpendicular to side A, C, such that point E, is above point A. Point B, lies on horizontal side A, C, point D, lies on hypotenuse C E, and line segment B D, is drawn such that it is parallel to side A, E, and forms triangle C D B. Side A, E, is labeled 18, side B C, is labeled 8, side B D, is labeled 6, and a right angle symbol is at point A

In the figure above, $line segment B D$ is parallel to $line segment A, E$ . What is the length of $line segment C E$ ?

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Explanation

The correct answer is 30. In the figure given, since $side B D$ is parallel to $side A, E$ and both segments are intersected by $side C E$ , then angle BDC and angle AEC are corresponding angles and therefore congruent. Angle BCD and angle ACE are also congruent because they are the same angle. Triangle BCD and triangle ACE are similar because if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. Since triangle BCD and triangle ACE are similar, their corresponding sides are proportional. So in triangle BCD and triangle ACE, $side B D$ corresponds to $side A, E$ and $side C D$ corresponds to $side C E$ . Therefore, $the length of side B D over the length of side C D, equals the length of side A, E over the length of side C E$ . Since triangle BCD is a right triangle, the Pythagorean theorem can be used to give the value of CD: $6 squared, plus 8 squared, equals the length of side C D squared$ . Taking the square root of each side gives $the length of side C D equals 10$ . Substituting the values in the proportion $the length of side B D over the length of side C D, equals the length of side A, E over the length of side C E$ yields $6 over 10, equals 18 over the length of C E$ . Multiplying each side by CE, and then multiplying by $10 over 6$ yields $the length of side C E equal to 30$ . Therefore, the length of $side C E$ is 30.