Two sides of a triangle each measure 4 mm. The angle between these two sides has a measure of...

Question:

Two sides of a triangle each measure 4 mm. The angle between these two sides has a measure of {eq}50^\circ. {/eq} What is the length of the third side of the triangle?

The Trigonometric Ratios:

The trigonometric ratios state the relationship between the three sides of a right triangle and one of its acute angles. The three basic trigonometric ratios are {eq}\rm sine(sin)\, cosine(cos)\, and\, tangent(tan) {/eq}.

Answer and Explanation:


The triangle described in our question is an isosceles triangle, with two of its legs measuring 4mm each. The angle at the apex of the triangle is 50 degrees.

Let the base of the isosceles triangle be {eq}x\rm\, mm {/eq}.

If we draw an altitude from the apex of the isosceles triangle to its base, it will divide the triangle into two congruent right triangles each with a hypotenuse of 4mm and a base of {eq}\dfrac{1}{2} x\, \rm mm {/eq}. The angle at the apex of the triangle, {eq}\angle 50^\circ {/eq} will also be divided into two.

Therefore, we can calculate the base of each of the right-angled triangle as:

  • {eq}\sin 25^\circ = \dfrac{\dfrac{1}{2} x\, \rm mm}{4\rm mm} {/eq}
  • {eq}\sin 25^\circ = \dfrac{1}{8} x {/eq}

Therefore:

  • {eq}x = 8\sin 25^\circ \approx \boxed{3.4\rm mm} {/eq}

Thus the third side of the triangle is 3.4 mm long.


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