Question

Which best describes the triangle or the triangles, if any, that can be formed with two sides that measure 5 inches and an angle that measures 60 degrees?
two different isosceles triangles
no triangle
one isosceles triangle
one equilateral triangle

Answer 1

Answer with explanation:

Suppose the two sides of triangle are, 5 inches and angle between the two sides is 60 degrees.

We will use cosine formula to find third side.Let side opposite to angle having measure 60 degree, is equal to a inches.

$cos 60^{\circ}=\frac{b^2+c^2-a^2}{2 b c}\\\\ \frac{1}{2}=\frac{5^2+5^2-a^2}{2\times 5 \times 5}\\\\25=50-a^2\\\\a^2=50-25\\\\a^2=25\\\\a=\pm 5$

Omitting, a= -5, because side can't be negative.

So, third side = 5 cm.

Means, triangle is equilateral.

If angle between , two sides having measure 5 cm , is not 60 degree, then the triangle will be always Isosceles.

we will sine rule to find remaining measurements.

$\frac{5}{\sin60^{\circ}}=\frac{5}{\sin B^{\circ}}=\frac{c}{\sin C^{\circ}}\\\\B=60^{\circ}\\\\ \text{by angle sum property of triangle}\\\\ C=60^{\circ}\\\\ \text{Side}\rightarrow c=5$

In this case,also ,triangle is equilateral.

Option D: one equilateral triangle

Answer 2

One isosceles triangle