Constructions and Loci

$\overline{AU,}\overline{BV}$ and $\overline{CW}$ are the medians of$\Delta ABC$

If ${\text{AP=x}}^{\text{2}}\text{ and PU}=\text{2x, then x=?}$

Given, $\Delta JKM$

Explain how to construct a triangle that is congruent to $\Delta JKM$.

Draw any $\overline{AB}$.

$\left(a\right).$ Construct $\overline{XY}$ 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 $\overline{XZ}$ and $\overline{YZ}$.

$\left(d\right).$ What kind of triangle is $\Delta XYZ?$

Explain how you could construct a $30°$ angle.

Exercise $3$ suggests that you could construct other angles with certain measures. Name some.

Suppose you are given the three lengths shown and are asked to construct a triangle whose sides have lengths $r,s,$and $t.$ Can you do so $?$ State the theorem from Chapter $6$ that applies.

$\angle 1$ and $\angle 2$ are given. You see two attempts at constructing an angle whose measure is equal to $m\angle 1+m\angle 2.$ Are both constructions satisfactory$?$

On your paper, draw two segments roughly like those shown. Use these segments in Exercise $1-4$ to construct a segment having the indicated length. $a+b$

On your paper, draw two segments roughly like those shown. Use these segments in Exercise $1-4$ to construct a segment having the indicated length. $b-a$

On your paper, draw two segments roughly like those shown. Use these segments in Exercise $1-4$ to construct a segment having the indicated length. $3a-b$

On your paper, draw two segments roughly like those shown. Use these segments in Exercise $1-4$ to construct a segment having the indicated length. $a+2b$

Using any convenient length for a side, construct an equilateral triangle.

$\left(a\right).$ Construct a $30°$ angle.

$\left(b\right).$ Construct a $15°$ angle.

$\left(a\right)$

Draw any acute $\Delta ACU$. Use a method based on the $SSS$ postulate to construct a triangle congruent to $\Delta ACU$.

Draw any obtuse $\Delta OBT$. Use the $SSS$ method to construct a triangle congruent to $\Delta OBT$.

Repeat Exercise $7$, but use the $SAS$ method.

Repeat Exercise $8$, but use the $ASA$ method.

On your paper, draw two angles roughly like those shown. Then for Exercise $11-14$ construct an angle having the indicated measure.  $x+y$

On your paper, draw two angles roughly like those shown. Then for Exercise $11-14$ construct an angle having the indicated measure.  $x-y$

On your paper, draw two angles roughly like those shown. Then for Exercise $11-14$ construct an angle having the indicated measure.  $\frac{3}{4}x$

On your paper, draw two angles roughly like those shown. Then for Exercise $11-14$ construct an angle having the indicated measure.  $180-2y$

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

$\left(b\right).$ 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, $\overrightarrow{AB}$. On $\overrightarrow{AB}$ mark off, in order, points $C$ and $D$ such that $AB=BC=CD$.

$\left(2\right).$ Construct equilateral $\Delta ADG$.

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

$\left(4\right).$ On $\overline{GD}$ 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 $\overleftrightarrow{FH},\overleftrightarrow{EB},$and $\overleftrightarrow{CI}$.

$\left(1\right)$

Construct an angle having the indicated measure. 

$120$

Construct an angle having the indicated measure. 

$150$

Construct an angle having the indicated measure. 

$165$

Construct an angle having the indicated measure. 

$45$

Draw any $\Delta ABC$. Construct $\Delta DEF$ so that $\Delta DEF~\Delta ABC$ and $DE=2AB$.

Construct a $\Delta RST$ such that $RS:ST:TR=4:6:7$.

On your paper draw figures roughly like those shown. Use them in constructing the figures described in Exercise $23-25$.

An isosceles triangle with a vertex angle of $n°$ and legs of length $d$.

On your paper draw figures roughly like those shown. Use them in constructing the figures described in Exercise $23-25$.

An isosceles triangle with a vertex angle of $n°$ and base of length $s$.

On your paper draw figures roughly like those shown. Use them in constructing the figures described in Exercise $23-25$.

An parallelogram with an $n°$ angle, longer side of length$s$ and longer diagonal of length $d$.

On your paper draw figures roughly like the ones shown. Then construct a triangle whose three angles are congruent to $\angle 1,\angle \text{2,}$ and $\angle 3$, and whose circumscribed circle has radius $r$.

A median of a triangle is a segment from a vertex to the _____ of the opposite side.

A quadrilateral with both pairs of opposite angles congruent is a ______ .

A parallelogram with congruent diagonal is a ______. 

A parallelogram with perpendicular diagonals is a ____.

If a side of a square has length $5cm$, then a diagonal of the square has length ____ cm.

The measure of each interior angle of a regular pentagon is  ____.

Suggest an alternate procedure for Construct $7$ that uses Constructions $5$ and $6$ .

Describe how you would construct each of the following:

The midpoint of $\overline{BC}$.

Describe how you would construct each of the following:

The median of $\Delta ABC$ that contains vertex $B$.

Describe how you would construct each of the following:

The altitude of $\Delta ABC$ that contain vertex$B$.

Describe how you would construct each of the following:

The altitude of $\Delta ABC$ that contain vertex$A$.

Describe how you would construct each of the following:

The perpendicular to $\overline{BC}$at $C$.

Describe how you would construct each of the following:

The square whose sides each have length$AC$.

Describe how you would construct each of the following:

A square whose perimeter equals $\overline{AC}$ .   

Describe how you would construct each of the following.

 A right triangle with hypotenuse and one leg equal to $AC$ and $BC$respectively.

The construction of the following triangle whose sides are in the ratio $1:2:\sqrt{5}$. 

Draw a figure roughly like one shown, but larger. Do the indicated construction clearly enough so that your method can be understood easily.