1

Problem

In triangle CDE, angle C measures 50°, angle D measures 70°, and side CE measures 10 m. Find the lengths of sides CD and DE.

2

Step

The given values are

C = 50°, D = 70° and CE = 10 m

3

Step

First, find angle E using the fact that the sum of angles in a triangle is 180°:

E = 180 - (C + D)

E = 180 - (50 + 70)

E = 60°

4

Formula

Now, let’s use the Law of Sines to find the length of side CD: \$(CD) / (sinC) = (DE)/ (sinD) = (CE) / (sinE)\$

5

Step

Plug the value in the formula and then simplify for the value of CD:

\$(CD) / (sin(50)) = (10)/ (sin(60))\$

\$CD = (10 \times sin(50)) / sin(60)\$

CD = 8.84 m

6

Formula

Now, let’s use the Law of Cosines to find the length of side DE: \$CosD = (CD)^2 + (CE)^2 - (DE)^2 / (2. CD. CE)\$.

7

Step

Rewrite the formula and then plug the value in the formula:

\$(DE)^2 = (CD)^2 + (CE)^2 - 2 \times CD \times CD \times cos(D)\$

\$(DE)^2 = (8.84)^2 + (10)^2 - 2 \times 8.84 times 10 \times cos(70)\$

8

Step

Simplify for the value of DE:

\$(DE)^2 = 78.14 + 100 - 176.4 \times 0.342\$

\$(DE)^2 = 117.674\$

\$DE = \sqrt (117.674)\$

DE = 10.84 m

9

Solution

So, the length of side CD is 8.84 m and side DE is 10.84 m.

10

Sumup

Please summarize Problem, Clue, Hint, Formula, Steps and Solution

Choices

11

Choice-A

Wrong: It suggests CD = 8.48 m and DE =10.84 m, which doesn’t match the calculations

Wrong CD = 8.48 m and DE = 10.84 m

12

Choice-B

Option B is incorrect because it doesn’t match the calculated values for both sides

Wrong CD = 8.48 m and DE = 10.48 m

13

Choice-C

Option C is correct because it matches these calculated values for CD and DE by using the law of sines and cosines formula

Correct CD = 8.84 m and DE = 10.84 m

14

Choice-D

Incorrect: It suggests that DE = 10.48m, which is not the correct value based on our calculations

Wrong CD = 8.84 m and DE = 10.48 m

15

Answer

Option

C

16

Sumup

Please summarize choices