Question

In ΔCDE, DE = 13, EC = 16, and CD = 7. Which list has the angles of ΔCDE in order from smallest to largest?

Answer 1

To determine the order of angles in ΔCDE from smallest to, we can use the Law of Cosines.

Let's label the angles as ∠C, ∠D, and ∠E, opposite to sides CD, DE, and EC, respectively.

Using the Law of Cosines, we have:

CD² = DE² + EC² - 2(DE)(EC)cos(∠C)

Substituting the given values, we get:

7² = 13² + 16² - 2(13)(16)cos(∠C)

Simplifying,

49 = 169 + 256 - 416cos(∠C)

Solving for cos(∠C),

cos(∠C) = (169 + 256 - 49) / (2(13)(16))
= 376 / 416
= 0.9038

Now, we can find the value of ∠C using the inverse cosine function:

∠C ≈ arccos(0.9038)
≈ 25.91°

Since ∠C is the smallest angle, it comes first in the list.

Next, we can find ∠D by applying the Law of Sines:

sin(∠D) / DE = sin(∠C) / CD

sin(∠D) = (DE * sin(∠C)) / CD
= (13 * sin(25.91°)) / 7
≈ 0.6937

∠D ≈ arcsin(0.6937)
≈ 44.66°

Finally, ∠E can be found by subtracting the sum of ∠C and ∠D from 180°:

∠E = 180° - ∠C - ∠D
≈ 180° - 25.91° - 44.66°
≈ 109.43°

Therefore, the order of angles in ΔCDE from smallest to largest is:
∠C ≈ 25.91°, ∠D ≈ 44.66°, ∠E ≈ 109.43°.

Answer 2

Explanation:

To determine the order of angles in ΔCDE from smallest to largest, we can use the Law of Cosines:

c^2 = a^2 + b^2 - 2ab cos(C)

where c is the side opposite angle C.

First, we can use this formula to find the measure of angle C:

7^2 = 13^2 + 16^2 - 2(13)(16) cos(C)

49 = 405 - 416 cos(C)

cos(C) = (405-49)/416 = 0.841

Using an inverse cosine function, we find that angle C is approximately 32.4 degrees.

Next, we can use the Law of Sines to find the measures of angles D and E:

(sin D) / 13 = (sin 32.4) / 7

sin D = 13 sin 32.4 / 7

D ≈ 61.4 degrees

(sin E) / 16 = (sin 32.4) / 7

sin E = 16 sin 32.4 / 7

E ≈ 86.6 degrees

Therefore, the angles of ΔCDE in order from smallest to largest are:

∠D ≈ 61.4°

∠C ≈ 32.4°

∠E ≈ 86.6°

So the answer is:

D. 61.4°, 32.4°, 86.6°