In right triangle ABC above, segment DE is drawn from side AB to AC as shown, forming right triangle ADE. If BC is 24, AB is 12, and BD is 4, what is the length of DE?
A.18
B.16
C.15
D.12
E.8
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16
To find the length of DE, we can use the concept of similar triangles. Triangle ADE is similar to triangle ABC, since angle A is shared by both triangles and angle D in triangle ADE is a right angle, just like angle C in triangle ABC.
Since the triangles are similar, we can set up a proportion to find the length of DE.
Let x be the length of DE.
From triangle ABC, we know that the ratio of the lengths of DE to BC is equal to the ratio of the lengths of AB to AC.
Therefore, we have:
DE/BC = AB/AC
Substituting the given values:
x/24 = 12/AC
To solve for x, we can cross-multiply:
x * AC = 12 * 24
x * AC = 288
Now, we need to find the length of AC. We can use the Pythagorean theorem in triangle ABC.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In triangle ABC, AC is the hypotenuse, and AB and BC are the other two sides.
Using the given values:
AC^2 = AB^2 + BC^2
AC^2 = 12^2 + 24^2
AC^2 = 144 + 576
AC^2 = 720
Now, take the square root of both sides to solve for AC:
AC = √720
AC ≈ 26.87
Substituting the value of AC back into the equation for x:
x * 26.87 = 288
x ≈ 288 / 26.87
x ≈ 10.72
Therefore, the length of DE is approximately 10.72.