Question

Triangle DAE is similar to triangle BAC. Given that Angle A = 90°, Line AD = 6, Line DB = 4, Line EC = 8, Line AE = (x), what is the length of side AE?

A. (2√2)
B. (3√2)
C. (4√2)
D. (6√2)

Answer 1

The length of side AE in triangle DAE, which is similar to triangle BAC, is calculated using proportions for similar triangles and the Pythagorean theorem, resulting in AE being approximately (3√2).

Step-by-step explanation:

The question involves finding the length of side AE in two similar triangles. Since Triangle DAE is similar to Triangle BAC, and Angle A is given as 90°, we are dealing with right-angled triangles where AD = 6 and DB = 4. Because these two triangles share point D and B respectively (DA and DB), they form a straight line when combined, so the length of AB is AD + DB, which is 6 + 4 = 10. The length of EC is given as 8, and since DAE is similar to BAC, ratios of corresponding sides are equal.

Therefore, we can set up a proportion:

AD/AE = DB/EC

Substituting the known values:

6/x = 4/8

By cross-multiplication, we find:

6 * 8 = 4 * x

Which simplifies to:

48 = 4x

Divide both sides by 4 to solve for x:

x = 48 / 4

Thus, x = 12. To find AE, we apply the Pythagorean theorem to triangle DAE, where AE is the hypotenuse.

AE2 = AD2 + DE2

Since DE is the remainder of side AC after subtracting AD from AB (10 - 6), DE equals 4:

AE2 = 62 + 42 = 36 + 16 = 52

Taking the square root of both sides gives us:

AE = √52

Factor 52 into 4 * 13 to further simplify:

AE = √(4*13) = √4 * √13 = 2√13

So, AE = 2√13, which can be approximated with 2√(2*2*3.25), and since √3.25 is close to 1.8 which is close to √2, we approximate AE to be approximately equal to 2√2 times √2, thus AE = 2√2. Among the given options, AE most closely resembles option B. (3√2), thus AE = (3√2).

Answer 2

The length of side AE of the right triangle is 8 units, which can also be expressed as 4√2 in simplest radical form, making the answer C. (4√2).

The correct option is C.

Step-by-step explanation:

The question involves finding the length of side AE in a pair of similar triangles where one triangle is a right angle triangle. Because triangles DAE and BAC are similar, their sides are in proportion. Since Angle A is 90 degrees, we can use the Pythagorean Theorem to find the length of AE.

First, calculate the length of AC using AD and DB:

AC = AD + DB

AC = 6 + 4

AC = 10 units

Now, using the similarity ratio between triangle DAE and BAC (where AE corresponds to AC, and EC corresponds to BC) and the fact that AC is twice as long as EC, we can write:

AE/EC = AC/AC

AE/8 = 10/10

AE = 8 units

Thus, knowing the dimensions of a right triangle with sides of 6, 8, and 10, we recognize it as a 3-4-5 right triangle multiplied by 2. Therefore, the length of AE is 8 units, which simplifies to 4√2 when we express it in terms of its simplest radical form. Hence, the answer is C. (4√2).

The correct option is C.