Question

5 from unit 1, lesson 5 noah is trying to bisect angle bac. he draws circles of the same radius with centers b and c and then uses one of the points of intersection for his ray as shown in this diagram. what mistake has noah made in his construction? 6 from unit 1, lesson 4 here is a straightedge and compass construction. use a straightedge to draw an equilateral triangle on the figure. explain how you know the triangle is equilateral.

Explanation:

Step1: Recall angle - bisection steps

To bisect an angle $\angle BAC$, circles should be drawn with centers on the rays $AB$ and $AC$ (not on points $B$ and $C$ which are not on the rays starting from $A$ of the angle to be bisected) with the same radius. Then the intersection of the circles is used to draw the bisecting ray.

Step2: Analyze Noah's construction

Noah drew circles with centers $B$ and $C$ instead of on the rays $AB$ and $AC$ of the angle $\angle BAC$ he wants to bisect. This will not result in a correct angle - bisecting ray.

Step3: Equilateral - triangle construction

For the equilateral - triangle construction, we can choose three non - collinear points among the intersection points of the circles. Since the circles are drawn with the same radius, if we consider the distance between any two intersection points of the circles, they are equal to the radius of the circles.
Let's say we choose three intersection points $P$, $Q$, $R$. The distance between $P$ and $Q$, $Q$ and $R$, $R$ and $P$ is equal to the radius of the circles used in the construction. By the definition of an equilateral triangle (a triangle with all sides equal), the triangle formed by these three points is equilateral.

Answer:

1. Noah's mistake: He drew circles with centers at $B$ and $C$ instead of on the rays $AB$ and $AC$ of the angle $\angle BAC$ he wants to bisect.
2. To draw an equilateral triangle: Select three non - collinear intersection points of the circles and use a straightedge to connect them. We know it is equilateral because the lengths of the sides of the triangle formed by these intersection points are equal to the radius of the circles used in the construction, so all three sides are equal.