Geometric Constructions and Theorems for Trigonometry

Practice fill-in-the-blank flashcards covering classical geometric constructions (line bisection, right angle, and inscribed polygons) and geometric theorems used in trigonometry.

The review of geometric constructions and theorems begins with concepts developed by the second-century astronomer and mathematician __________.

Ptolemy

To bisect line $AB$, circles are drawn with centers at $A$ and $B$ and radii $AB$ and $BA$, intersecting at points $C$ and $D$; drawing line $CD$ determines point $E$, which __________ line $AB$.

bisects

To construct a right angle at point $A$ on line $AB$, describe a circle with center at random point $C$ (not on $AB$) and radius $CA$, draw diameter $ECD$, and draw line $AD$ to form right angle __________.

$$\angle DAB$$

The first step to inscribe an equilateral triangle in circle $ABC$ is to draw diameter __________.

$$ADE$$

When inscribing an equilateral triangle in circle $ABC$, after drawing diameter $ADE$, draw a circle with center $E$ and radius __________ to determine points on the circle.

$$ED$$

To inscribe a square in circle $ABCD$, draw diameter $BD$, and then at center $E$, draw line $AC$ at __________ angles to $BD$.

right

In the square inscription construction, drawing lines $AB$, $BC$, $CD$, and $DA$ forms square __________ inscribed in circle $ABCD$.

$$ABCD$$

To inscribe an equilateral pentagon in circle $CNMKP$, first draw diameter $CBA$, and then at point $A$, draw line __________ at right angles to $CBA$.

$$JD$$

In the pentagon inscription procedure, drawing a circle with center $E$ and radius $EA$ determines points $F$ and $G$, after which drawing line $FG$ determines point __________.

$$H$$

In the equilateral pentagon inscription, step 6 requires drawing a circle with center $H$ and radius __________ to determine point $J$.

$$HB$$

The final step of the pentagon construction joins lines $KM$, $MN$, $NC$, $CP$, and $PK$ to form equilateral pentagon __________ in circle $CNMKP$.

$$KMNCP$$

To inscribe an equilateral hexagon in circle $ABC$, first draw diameter $ADE$, then draw a circle with center $E$ and radius __________.

$$ED$$

When inscribing an equilateral hexagon in circle $ABC$, line $BD$ is drawn and produced to $F$, and line $CD$ is drawn and produced to __________.

$$G$$

Joining lines $AG$, $GB$, $BE$, $EC$, $CF$, and $FA$ forms equilateral hexagon __________ inscribed in circle $ABC$.

$$AGBECF$$

In the problem set, all regular polygon inscriptions (triangle, square, pentagon, and hexagon) are constructed in a circle having a radius of __________.

$$4\,\text{cm}$$

The Equilateral Triangle Theorem states that if an equilateral triangle is inscribed in a circle, the square on the side of the equilateral triangle is __________ the square on the radius.

triple

In circle $ABC$ with diameter $ADE$, the algebraic conclusion proven by the Equilateral Triangle Theorem is $AB^2 =$ __________.

$$3AD^2$$

In the proof of the Equilateral Triangle Theorem, the step stating $AE^2 = AB^2 + BE^2$ is justified by the __________ Theorem.

Pythagorean

The Square Theorem proves that a square inscribed in a circle is twice the square on the radius, which is expressed algebraically as $AB^2 =$ __________.

$$2AE^2$$