91Ó°ÊÓ

Draw, if possible, a triangle in which the perpendicular bisectors of the sides intersect in a point with the location described.

1. A point inside the triangle
2. A point outside the triangle
3. A point on the triangle

$\left(a\right).$Draw any acute triangle. Bisect each of the three angles.

role="math" localid="1648851512702" $\left(b\right).$Draw any obtuse triangle. Bisect each of the three angles

$\left(c\right).$What do you notice about the points of intersection of the bisectors in parts $\left(a\right)$ and $\left(b\right)?$

$\left(a\right)$

Construct a six-pointed star using the following procedure.

$\left(1\right).$ Draw a ray, $\stackrel{→}{AB}$. On $\stackrel{→}{AB}$ mark off, in order, points $C$ and $D$ such that $AB=BC=CD$.

$\left(2\right).$ Construct equilateral $\mathrm{Δ}ADG$.

$\left(3\right).$ On $\stackrel{¯}{AG}$ mark off points $E$and $F$ so that both $AE$and $EF$ equal $AB$.

$\left(4\right).$ On $\stackrel{¯}{GD}$ mark off points $H$and $I$ so that both $GH$and $HI$ equal $AB$.

$\left(5\right).$To complete the star, draw the three lines $\stackrel{↔}{FH},\stackrel{↔}{EB},$and $\stackrel{↔}{CI}$.

$\left(1\right)$

Draw any $\stackrel{¯}{AB}$.

$\left(a\right).$ Construct $\stackrel{¯}{XY}$ so that $XY=AB$.

$\left(b\right).$ Using $X$ and $Y$ as centers, and a radius equal to $AB,$ draw arcs that intersect. Label the point of intersection $Z$.

$\left(c\right).$ Draw $\stackrel{¯}{XZ}$ and $\stackrel{¯}{YZ}$.

$\left(d\right).$ What kind of triangle is $\mathrm{Δ}XYZ?$

Draw a diagram roughly like the one shown. Without laying your straightedge across any part of the lake, construct more of $\stackrel{→}{RS}$.

Draw three noncollinear points $R,S,$and $T.$ Construct a triangle whose sides have $R,S,$and $T$ as midpoints.

Draw a segment and let its length be $1$.

$\left(a\right).$ Construct a segment of length $\sqrt{5}$.

$\left(b\right).$ Construct a segment of length $\frac{1}{2}+\frac{\sqrt{5}}{2}$, or $\frac{1+\sqrt{5}}{2}$.

$\left(c\right).$ Construct a golden rectangle whose sides are in the ratio $1:\frac{1+\sqrt{5}}{2}$.

$\left(a\right)$

Is there some kind of triangle such that the perpendicular bisector of each side is also an angle bisector, a median, and an altitude?

A parallelogram with perpendicular diagonals is a ____.

Use a ruler and a protractor to draw a regular pentagon. Draw a regular pentagon as in before exercise. Construct the perpendicular bisectors.