9. Polygons & Quadrilaterals

New York Geometry - 2020 Edition

9.07 Justifying constructions using the properties of quadrilaterals

Lesson

We can use a set of circular arcs to construct geometric figures like quadrilaterals. We can use the property of quadrilaterals to go one step further and construct bisectors, angle bisectors, or a pair of perpendicular or parallel lines. For instance, the following pair of perpendicular line segments are the diagonals of the rhombus $ACBD$`A``C``B``D`.

$\overline{AB}$AB is perpendicular to $\overline{DC}$DC |

So given a set of arcs, we want to identify what quadrilateral was made in order to prove geometric statements.

Let's determine which order the arcs were drawn. From the image above, we can see that we are given the point $A$`A`, and that the arcs below were drawn first.

First, arcs drawn with center at $A$A |

This allows us to place the point $B$`B` along the middle arc, and create the second pair of arcs with same radius to create the points $C$`C` and $D$`D`.

Then, arcs were drawn centered at $B$B |

Then the perpendicular lines were drawn.

$\overline{AB}$AB is perpendicular to $\overline{DC}$DC |

To prove that $\overline{AB}$`A``B` is perpendicular to $\overline{DC}$`D``C`, we first notice that the sides $\overline{AC}$`A``C`, $\overline{CB}$`C``B`, $\overline{BD}$`B``D`, and $\overline{DA}$`D``A` are all congruent since they've been constructed by the same radius. So by definition, this means that $ACBD$`A``C``B``D` is a rhombus. A special property of a rhombus is that the diagonals are perpendicular. In this case, $\overline{AB}$`A``B` is perpendicular to $\overline{DC}$`D``C`.

We can formalize the above explanation in a two column proof as shown below.

Statements |
Reasons |

$\overline{AC}\cong\overline{CB}\cong\overline{BD}\cong\overline{DA}$ |
The arcs that created the segments have the same radius. |

$ACBD$ACBD is a rhombus |
Definition of a rhombus |

$\overline{AB}\cong\overline{DC}$AB≅DC |
If a parallelogram is a rhombus, then its diagonals are perpendicular. |

In the image below, the line $\overleftrightarrow{AB}$›‹`A``B` is constructed parallel to the given line $\overleftrightarrow{DC}$›‹`D``C`. Justify the steps of construction in a two column proof.

In the image below, $\overline{PR}$`P``R` is constructed so that $\angle TPR$∠`T``P``R` is congruent to $\angle RPQ$∠`R``P``Q`. Justify the steps of construction in a two column proof.

Make, justify, and apply formal geometric constructions.