Math Problem Statement

In ΔDEF, e = 1.8 inches, mm∠F=104° and mm∠D=52°. Find the length of f, to the nearest 10th of an inch.

Solution

In triangle ΔDEF\Delta DEF, we know:

  • Side e=1.8e = 1.8 inches,
  • mF=104m\angle F = 104^\circ,
  • mD=52m\angle D = 52^\circ.

To find side ff, let's proceed with the following steps:

Step 1: Calculate mEm\angle E

Since the sum of angles in a triangle is 180180^\circ, mE=180mFmDm\angle E = 180^\circ - m\angle F - m\angle D mE=18010452=24m\angle E = 180^\circ - 104^\circ - 52^\circ = 24^\circ

Step 2: Use the Law of Sines

The Law of Sines states: esin(D)=fsin(E)\frac{e}{\sin(D)} = \frac{f}{\sin(E)}

Rearrange to solve for ff: f=esin(E)sin(D)f = e \cdot \frac{\sin(E)}{\sin(D)}

Substitute the values: f=1.8sin(24)sin(52)f = 1.8 \cdot \frac{\sin(24^\circ)}{\sin(52^\circ)}

Step 3: Calculate ff (using approximate sine values)

Using approximate values, sin(24)0.4067\sin(24^\circ) \approx 0.4067 and sin(52)0.7880\sin(52^\circ) \approx 0.7880, f=1.80.40670.78801.80.5160.9f = 1.8 \cdot \frac{0.4067}{0.7880} \approx 1.8 \cdot 0.516 \approx 0.9

Answer:

The length of ff is approximately 0.9 inches to the nearest tenth.

Would you like more details on any of the steps?


Here are some additional questions to deepen understanding:

  1. How would the calculation change if angle FF were an acute angle?
  2. What if side ee were doubled? How would that affect ff?
  3. How would using the Law of Cosines help in a scenario where we have different known values?
  4. Can we solve for dd in the triangle, given the current information?
  5. How does changing angle DD impact the overall shape and measurements of the triangle?

Tip: Remember that the Law of Sines is particularly useful when you know two angles and one side or two sides and a non-included angle.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Triangle Properties
Angle Sum in Triangle
Law of Sines

Formulas

m∠E = 180° - m∠F - m∠D
Law of Sines: e / sin(D) = f / sin(E)
f = e * (sin(E) / sin(D))

Theorems

Law of Sines
Triangle Angle Sum Theorem

Suitable Grade Level

Grades 10-12