Math Problem Statement

Solve for x in triangle DEF with sides DE = 2x, DF = 7, EF = x + 1, and angle E = 60 degrees.

Solution

The triangle provided has the following properties:

  • Side DE=2xDE = 2x
  • Side DF=7DF = 7
  • Side EF=x+1EF = x + 1
  • Angle E=60\angle E = 60^\circ

We can solve for the unknowns in this triangle using the Law of Cosines, which is useful for non-right triangles.

Law of Cosines:

c2=a2+b22abcos(C)c^2 = a^2 + b^2 - 2ab \cdot \cos(C) Where aa, bb, and cc are the sides of the triangle, and CC is the included angle between sides aa and bb.

For this triangle:

  • a=7a = 7 (side DFDF)
  • b=x+1b = x + 1 (side EFEF)
  • c=2xc = 2x (side DEDE)
  • C=60C = 60^\circ

Substitute into the Law of Cosines formula: (2x)2=72+(x+1)227(x+1)cos(60)(2x)^2 = 7^2 + (x + 1)^2 - 2 \cdot 7 \cdot (x + 1) \cdot \cos(60^\circ) Since cos(60)=0.5\cos(60^\circ) = 0.5, the equation simplifies to: 4x2=49+(x+1)27(x+1)4x^2 = 49 + (x + 1)^2 - 7 \cdot (x + 1)

Now, solve for xx. Let me calculate that.The solutions for xx are:

x=5+5416orx=55416x = \frac{-5 + \sqrt{541}}{6} \quad \text{or} \quad x = \frac{-5 - \sqrt{541}}{6}

Since xx represents a length in the triangle, only the positive solution is valid: x=5+54163.027x = \frac{-5 + \sqrt{541}}{6} \approx 3.027

Thus, x3.03x \approx 3.03.

Would you like further details or have any questions?

Here are some follow-up questions to deepen your understanding:

  1. What is the value of the side DEDE now that we know xx?
  2. How can the Law of Sines be applied to check the consistency of this solution?
  3. How would this problem change if the angle was different from 6060^\circ?
  4. Can you compute the area of the triangle using Heron's formula with these side lengths?
  5. What other types of triangles can we apply the Law of Cosines to?

Tip: The Law of Cosines is especially useful when you know two sides and the included angle, or when you know all three sides but need to find an angle.

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Math Problem Analysis

Mathematical Concepts

Geometry
Trigonometry

Formulas

Law of Cosines: c^2 = a^2 + b^2 - 2ab * cos(C)

Theorems

Law of Cosines

Suitable Grade Level

Grades 9-11