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Geometry and Trigonometry Difficulty: Hard

In the figure shown, $upper A upper B equals StartRoot 34 EndRoot$ units, $upper A upper C equals 3$ units, and $upper C upper E equals 21$ units. What is the area, in square units, of triangle $A D E$ ?

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Explanation

The correct answer is $480$ . It's given in the figure that angle $A C B$ and angle $A E D$ are right angles. It follows that angle $A C B$ is congruent to angle $A E D$ . It's also given that angle $B A C$ and angle $D A E$ are the same angle. It follows that angle $B A C$ is congruent to angle $D A E$ . Since triangles $A B C$ and $A D E$ have two pairs of congruent angles, the triangles are similar. Sides $A B$ and $A C$ in triangle $A B C$ correspond to sides $A D$ and $A E$ , respectively, in triangle $A D E$ . Corresponding sides in similar triangles are proportional. Therefore, $\frac{A D}{A B} = \frac{A E}{A C}$ . It's given that $upper A upper C equals 3$ units and $upper C upper E equals 21$ units. Therefore, $upper A upper E equals 24$ units. It’s also given that $upper A upper B equals StartRoot 34 EndRoot$ units. Substituting $3$ for $A C$ , $24$ for $A E$ , and $StartRoot 34 EndRoot$ for $A B$ in the equation $\frac{A D}{A B} = \frac{A E}{A C}$ yields $\frac{A D}{\sqrt{34}} = \frac{24}{3}$ , or $\frac{A D}{\sqrt{34}} = 8$ . Multiplying each side of this equation by $StartRoot 34 EndRoot$ yields $upper A upper D equals 8 StartRoot 34 EndRoot$ . By the Pythagorean theorem, if a right triangle has a hypotenuse with length $c$ and legs with lengths $a$ and $b$ , then $a^{2} + b^{2} = c^{2}$ . Since triangle $A D E$ is a right triangle, it follows that $A D$ represents the length of the hypotenuse, $c$ , and $D E$ and $A E$ represent the lengths of the legs, $a$ and $b$ . Substituting $24$ for $b$ and $8 StartRoot 34 EndRoot$ for $c$ in the equation $a^{2} + b^{2} = c^{2}$ yields $a^{2} + {(24)}^{2} = {(8 \sqrt{34})}^{2}$ , which is equivalent to $a^{2} + 576 = 64 (34)$ , or $a^{2} + 576 = 2,176$ . Subtracting $576$ from both sides of this equation yields $a squared equals 1,600$ . Taking the square root of both sides of this equation yields $a = \pm 40$ . Since $a$ represents a length, which must be positive, the value of $a$ is $40$ . Therefore, $upper D upper E equals 40$ . Since $D E$ and $A E$ represent the lengths of the legs of triangle $A D E$ , it follows that $D E$ and $A E$ can be used to calculate the area, in square units, of the triangle as $one half left parenthesis 40 right parenthesis left parenthesis 24 right parenthesis$ , or $480$ . Therefore, the area, in square units, of triangle $A D E$ is $480$ .