Question

lesson 6 practice problems

1. which of the following constructions would help to construct a line passing through point c that is perpendicular to the line ab?

a. construction of an equilateral triangle with one side ab
b. construction of a hexagon with one side bc
c. construction of a perpendicular bisector through c
d. construction of a square with one side ab

1. two distinct lines, ℓ and m, are each perpendicular to the same line n. select all the true statements.

a. lines ℓ and m are perpendicular.
b. lines ℓ and n are perpendicular.
c. lines m and n are perpendicular.
d. lines ℓ and m are parallel.
e. lines ℓ and n are parallel.
f. lines m and n are parallel.

1. this diagram is a straightedge and compass construction of the bisector of angle bac. only angle bac is given. explain the steps of the construction in order. include a step for each new circle, line, and point.

Explanation:

Step1: Analyze question 1

The construction of a perpendicular bisector through a point to a line will give a line perpendicular to the given line passing through that point. Options A, B and D do not directly help in constructing a line through C perpendicular to AB.

Step2: Analyze question 2

If two distinct lines $\ell$ and $m$ are each perpendicular to the same line $n$, then by the perpendicular - parallel relationship in a plane, $\ell$ and $m$ are parallel, and $\ell$ and $n$ are perpendicular, $m$ and $n$ are perpendicular.

Step3: Analyze question 3

1. Place the compass at point A. Draw a circle that intersects rays AB and AC at two distinct points, say B and C.
2. Place the compass at point B. Draw a circle with a radius greater than half the distance between the intersection points of the first - circle and AB and AC.
3. Without changing the compass width, place the compass at point C and draw another circle. These two circles intersect at two points, say D and E.
4. Draw the line DE. The line DE is the angle - bisector of $\angle BAC$.

Answer:

1. C
2. B, C, D
3. 1. Place the compass at point A. Draw a circle that intersects rays AB and AC at two distinct points.
4. Place the compass at one of the intersection points (e.g., B) on AB. Draw a circle with a suitable radius.
5. Place the compass at the other intersection point (e.g., C) on AC. Draw a circle with the same radius as in step 2.
6. The two new - drawn circles intersect at two points. Draw the line through these two intersection points. This line is the bisector of $\angle BAC$.