Question

5
multiple choice 5 points
which statement about two equilateral triangles is always true?
they are congruent.
they are equal in area.
they have congruent altitudes.
they are similar.

Explanation:

Step1: Analyze the properties of congruent triangles

Congruent triangles have all corresponding sides and angles equal. Two equilateral triangles can have different side lengths. For example, one with side length \(a = 2\) and another with side length \(b=3\). So, they are not always congruent.

Step2: Analyze the formula for the area of an equilateral triangle

The area of an equilateral triangle is \(A=\frac{\sqrt{3}}{4}s^{2}\), where \(s\) is the side - length. If \(s_1
eq s_2\), then \(A_1=\frac{\sqrt{3}}{4}s_1^{2}
eq A_2=\frac{\sqrt{3}}{4}s_2^{2}\). So, they are not always equal in area.

Step3: Analyze the formula for the altitude of an equilateral triangle

The altitude \(h\) of an equilateral triangle is \(h = \frac{\sqrt{3}}{2}s\). If \(s_1
eq s_2\), then \(h_1=\frac{\sqrt{3}}{2}s_1
eq h_2=\frac{\sqrt{3}}{2}s_2\). So, they do not always have congruent altitudes.

Step4: Analyze the similarity of equilateral triangles

For two triangles to be similar, the ratios of their corresponding side lengths must be equal and their corresponding angles must be equal. In an equilateral triangle, all angles are \(60^{\circ}\). Let the side lengths of two equilateral triangles be \(s_1\) and \(s_2\). The ratio of their side lengths is \(\frac{s_1}{s_2}\), and all corresponding angles are \(60^{\circ}\). By the AA (angle - angle) similarity criterion (since all three angles of one equilateral triangle are equal to all three angles of another equilateral triangle), two equilateral triangles are always similar.

Answer:

They are similar.