Mix Examples - Constructions Questions in English

(N/A) Steps of construction:
$1$. Draw a circle with center $O$ and radius $4 \text{ cm}$.
$2$. Mark a point $X$ outside the circle such that the distance $OX = 7 \text{ cm}$.
$3$. Draw the perpendicular bisector of the line segment $OX$. Let $M$ be the midpoint of $OX$.
$4$. With $M$ as the center and $MO$ as the radius,draw a circle. This circle will intersect the original circle at two points,say $P$ and $Q$.
$5$. Join $XP$ and $XQ$. These are the required tangents to the circle from point $X$.

(N/A) Steps of construction:
$1$. Draw a circle with center $P$ and radius $5 \,cm$.
$2$. Mark a point $A$ outside the circle such that the distance $PA = 9 \,cm$.
$3$. Draw the perpendicular bisector of the line segment $PA$. Let $M$ be the midpoint of $PA$.
$4$. With $M$ as the center and $MA$ (or $MP$) as the radius,draw a circle. This circle will intersect the original circle at two points,say $B$ and $C$.
$5$. Join $AB$ and $AC$. Thus,$AB$ and $AC$ are the required pair of tangents to the circle from point $A$.

(N/A) Step $1$: Place a round bowl upside-down on a paper and trace its boundary to draw a circle.
Step $2$: Since the center of the circle is unknown,draw two non-parallel chords $AB$ and $CD$ of the circle.
Step $3$: Draw the perpendicular bisectors of chords $AB$ and $CD$. The point where these bisectors intersect is the center $O$ of the circle.
Step $4$: Mark a point $P$ outside the circle.
Step $5$: Join $OP$ and draw the perpendicular bisector of $OP$. Let $M$ be the midpoint of $OP$.
Step $6$: With $M$ as the center and $MO$ as the radius,draw a circle. Let this circle intersect the original circle at points $Q$ and $R$.
Step $7$: Join $PQ$ and $PR$. $PQ$ and $PR$ are the required tangents to the circle from point $P$.

(N/A) Steps of construction:
$1$. Draw a circle with center $O$ and radius $4\, cm$.
$2$. Draw a diameter $\overline{PQ}$ passing through the center $O$.
$3$. Extend the diameter $\overline{PQ}$ on both sides to locate points $A$ and $B$ such that $A-P-Q$ and $P-Q-B$. Let $OA = OB = 6\, cm$ (or any distance greater than the radius).
$4$. Find the midpoints of $\overline{OA}$ and $\overline{OB}$ by constructing perpendicular bisectors. Let these midpoints be $M_1$ and $M_2$ respectively.
$5$. With $M_1$ as center and $M_1O$ as radius,draw a circle. It intersects the original circle at two points. Join $A$ to these points to get the tangents from $A$.
$6$. Similarly,with $M_2$ as center and $M_2O$ as radius,draw a circle. It intersects the original circle at two points. Join $B$ to these points to get the tangents from $B$.

(N/A) Steps of construction:
$1$. Draw a circle with center $P$ and radius $3 \, cm$.
$2$. Draw a diameter $\overline{ AB }$ passing through $P$.
$3$. Extend the line segment $\overline{ AB }$ on both sides. Mark point $X$ on the side of $A$ such that $PX = 7 \, cm$ and point $Y$ on the side of $B$ such that $PY = 7 \, cm$.
$4$. To draw tangents from $X$,find the midpoint $M_1$ of $\overline{ PX }$. With $M_1$ as center and $M_1P$ as radius,draw a circle. Let it intersect the original circle at points $Q$ and $R$.
$5$. Join $XQ$ and $XR$. These are the required tangents from $X$.
$6$. Similarly,to draw tangents from $Y$,find the midpoint $M_2$ of $\overline{ PY }$. With $M_2$ as center and $M_2P$ as radius,draw a circle. Let it intersect the original circle at points $S$ and $T$.
$7$. Join $YS$ and $YT$. These are the required tangents from $Y$.

(N/A) Steps of construction:
$1$. Draw a line segment $\overline{ AB } = 9\, cm$.
$2$. With $A$ as the center,draw a circle of radius $3\, cm$. With $B$ as the center,draw a circle of radius $3.5\, cm$.
$3$. To draw tangents from $B$ to $\odot( A, 3\, cm )$:
a. Find the midpoint $M$ of $\overline{ AB }$ by drawing the perpendicular bisector of $\overline{ AB }$.
b. With $M$ as the center and $MA$ as the radius,draw a circle. This circle intersects $\odot( A, 3\, cm )$ at points $P$ and $Q$.
c. Join $BP$ and $BQ$. These are the required tangents from $B$ to the circle centered at $A$.
$4$. To draw tangents from $A$ to $\odot( B, 3.5\, cm )$:
a. With $M$ as the center and $MB$ as the radius,draw a circle (which is the same as the one drawn in step $3b$). This circle intersects $\odot( B, 3.5\, cm )$ at points $R$ and $S$.
b. Join $AR$ and $AS$. These are the required tangents from $A$ to the circle centered at $B$.

(N/A) Steps of construction:
$1$. Draw a line segment $\overline{ XY } = 11\, cm$.
$2$. Draw a circle with center $X$ and radius $4\, cm$,and another circle with center $Y$ and radius $3\, cm$.
$3$. Find the midpoint $M$ of $\overline{ XY }$ by drawing the perpendicular bisector of $\overline{ XY }$.
$4$. With $M$ as the center and $MX$ (or $MY$) as the radius,draw a circle. This circle will intersect the circle with center $X$ at points $A$ and $B$,and the circle with center $Y$ at points $C$ and $D$.
$5$. Join $XA$ and $XB$ to get the tangents from $X$ to the circle centered at $Y$. (Note: Since the circles are separate,tangents from the center of one circle to the other are drawn by finding the intersection points of the auxiliary circle with the target circle).
$6$. Join $YC$ and $YD$ to get the tangents from $Y$ to the circle centered at $X$.

(N/A) $1$. Draw a circle with center $P$ and radius $5 \, cm$.
$2$. Draw a radius $PA$ and another radius $PB$ such that $\angle APB = 120^\circ$ (since the angle between tangents is $60^\circ$,the angle at the center is $180^\circ - 60^\circ = 120^\circ$).
$3$. At point $A$,construct a line perpendicular to $PA$.
$4$. At point $B$,construct a line perpendicular to $PB$.
$5$. Let these two perpendicular lines intersect at point $Q$. Then $QA$ and $QB$ are the required tangents,and $\angle AQB = 60^\circ$.

(N/A) $1$. Draw a circle with center $P$ and radius $4\, cm$.
$2$. Draw two perpendicular diameters $AB$ and $CD$ of the circle,intersecting at the center $P$.
$3$. At points $A, B, C,$ and $D$,draw tangents to the circle using a set square or by constructing $90^{\circ}$ angles at each point.
$4$. Let the tangents at $A$ and $C$ intersect at point $Q$. Since the radii $PA$ and $PC$ are perpendicular,the quadrilateral $PAQC$ is a square (as all angles are $90^{\circ}$ and adjacent sides are equal to the radius).
$5$. Thus,the tangents at $A$ and $C$ are perpendicular to each other at point $Q$.

(N/A) $1$. Draw a circle with center $P$ and radius $5 \, cm$.
$2$. Draw a radius $PA$ and another radius $PB$ such that $\angle APB = 180^\circ - 120^\circ = 60^\circ$. This is because the angle between the tangents and the angle between the radii at the points of contact are supplementary $(180^\circ)$.
$3$. At point $A$,construct a line perpendicular to $PA$.
$4$. At point $B$,construct a line perpendicular to $PB$.
$5$. Let these two perpendicular lines intersect at point $Q$. The lines $QA$ and $QB$ are the required tangents,and $\angle AQB = 120^\circ$.