Question

materials: compass; blank paper, a ruler, and a protractor.
a) using the given line segment ab, and taking segment ab as one side, draw and equilateral triangle abc. what can you say about the length of the remaining two sides?
b) with segment ab as one side, draw an isosceles triangle. what is the length of each remaining side?

Explanation:

Step1: Recall properties of equilateral triangle

In an equilateral triangle, all sides are equal. Given side AB, if $\triangle ABC$ is equilateral, then $AB = BC=AC$.

Step2: Recall properties of isosceles triangle

An isosceles triangle has two equal - sides. If side AB is one side of the isosceles triangle, let the length of AB be $l$. If AB is the base, the lengths of the other two sides (the legs) can be any length greater than $\frac{l}{2}$ (by the triangle - inequality theorem). If AB is one of the equal sides, then one of the remaining sides has length $l$ and the other side can have any positive length less than $2l$ (again, by the triangle - inequality theorem).

Answer:

a) The lengths of the remaining two sides (BC and AC) are equal to the length of side AB.
b) If AB is the base, the lengths of the two equal sides can be any length greater than half of the length of AB. If AB is one of the equal sides, one of the remaining sides has the same length as AB and the other side has a length greater than 0 and less than twice the length of AB.