Question

marco wants to draw a triangle, abc, where the measure of ∠a is 60°, the measure of ab is 5 inches, and the measure of bc is 5 inches. which statement is true about the triangle marco wants to draw?
triangle abc will be isosceles but not equilateral.
triangle abc will be scalene.
it is not possible to draw this triangle.
triangle abc will be equilateral.

Explanation:

Step1: Recall triangle - side and angle properties

In \(\triangle ABC\), \(AB = 5\) inches, \(BC = 5\) inches, and \(\angle A=60^{\circ}\). Since two sides are equal (\(AB = BC\)), the triangle is isosceles.

Step2: Apply the Law of Cosines or angle - side relationships

Let \(AB = c = 5\), \(BC=a = 5\). Using the Law of Cosines \(a^{2}=b^{2}+c^{2}-2bc\cos A\). Substituting \(a = 5\), \(c = 5\) and \(\angle A = 60^{\circ}\) (\(\cos A=\frac{1}{2}\)):
\[

$$\begin{align*} 25&=b^{2}+25 - 2\times b\times5\times\frac{1}{2}\\ 25&=b^{2}+25 - 5b\\ b^{2}-5b&=0\\ b(b - 5)&=0 \end{align*}$$

\]
So \(b = 0\) (rejected) or \(b = 5\).
Since \(AB=BC = AC=5\) inches and \(\angle A = 60^{\circ}\), by the definition of an equilateral triangle (all sides equal and all angles \(60^{\circ}\)), the triangle is equilateral.

Answer:

Triangle \(ABC\) will be equilateral.