Angle upper D upper E upper A is a right angle. Point upper B lies on side upper A upper

Question

Angle upper D upper E upper A is a right angle.
Point upper B lies on side upper A upper D.
Point upper C lies on side upper A upper E.
Line segment upper B upper C is drawn from side upper A upper D to side upper A upper E to form right triangle upper A upper B upper C.
Angle upper B upper C upper A is a right angle.
A note indies the figure is not drawn to scale.

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 $A D E$ ?

Accepted Answer

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, A D A B = A E A C . It's given that A C = 3 units and C E = 21 units. Therefore, A E = 24 units. It&rsquo;s also given that A B = 34 units. Substituting 3 for A C , 24 for A E , and 34 for A B in the equation A D A B = A E A C yields A D 34 = 24 3 , or A D 34 = 8 . Multiplying each side of this equation by 34 yields A D = 8 34 . 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 34 for c in the equation a 2 + b 2 = c 2 yields a 2 + 24 2 = 8 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 2 = 1,600 . Taking the square root of both sides of this equation yields a = &#177; 40 . Since a represents a length, which must be positive, the value of a is 40 . Therefore, D E = 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 1 2 40 24 , or 480 . Therefore, the area, in square units, of triangle A D E is 480 .

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