Construction: Scaling a line, triangle, polygon, circle

overview

In this page, the following are covered

dividing a line in a given ratio (The method employed is based on properties of similar triangles.)

scaling or dividing a line in a given ratio (The method employed uses properties of traversal of pair of parallels.)

scaling a triangle in a given ratio

scaling a polygon in a given ratio

scaling a circle

problem

Given a line segment $\overline{AB}$, it is required to mark a point $P$ such that, $\overline{AP}:\overline{PB}=m:n$. That is, the $\overline{AB}$ is divided to $\overline{AP}$ and $\overline{PB}$ in $m:n$ ratio.

To achieve that, we use properties of similar triangles.

A set of similar triangles are formed with $\overline{AB}$ as one of its sides. The formulation is shown in the figure. In such a construction, the following properties are noted.

• $\overline{AB}:\overline{AP}=\overline{AD}:\overline{AC}$

• $\overline{EB}\mid \mid \overline{CP}$

The construction is explained in following pages.

The construction is visualized in the following.

• Construct a ray $\overrightarrow{AR}$ at *an angle*, of any measurement.

• In a compass, take a measurement

• Using the compass, mark a number of segments on the ray.

• Construct line $D$ to $B$

• Construct a line through the the point $C$

• The point of intersection of the line through $C$ and $\overline{AB}$ is marked $P$.

The figure illustrates $3$ different angles $\angle AR1$, $\angle AR2$, and $\angle AR3$. The procedure to construct will work for any arbitrary measure of angle.

Continuing on dividing line segment $\overline{AB}$ in ratio $\overline{AP}:\overline{PB}=2:1$. The construction is visualized in the following.

• Construct a ray $\overrightarrow{AR}$ at an angle

• In a compass, take *a measurement*. Any arbitrary length will do fine.

• Using the compass, mark a number of segments on the ray.

• Construct line $D$ to $B$

• Construct a line through the the point $C$

• The point of intersection of line through $C$ and $\overline{AB}$ is marked $P$.

Continuing on dividing $\overline{AB}$ in ratio $2:1$. The figure illustrates $3$ different measures on compass. The procedure to construct will work for any arbitrary length.

Continuing on dividing line segment $\overline{AB}$ in ratio $\overline{AP}:\overline{PB}=2:1$. The construction is visualized in the following.

• Construct a ray $\overrightarrow{AR}$ at an angle

• In a compass, take a measurement

• Using the compass, mark *a number of segments* on the ray. To achieve $2:1$, sum of $2$ and $1$ = $3$ segments are marked.,

• Construct line $D$ to $B$

• Construct a parallel to $\overline{DB}$ through the the point $C$

• The point of intersection of parallel through $C$ and $\overline{AB}$ is marked $P$.

Continuing on dividing $\overline{AB}$ in ratio $2:1$. The lines through points $C1$, $C2$, and $C3$ are parallel to $\overline{D1B}$, $\overline{D2B}$, and $\overline{D3B}$ respectively. By the rules of similar triangles, the lines intersect at point $P$.

The construction is visualized in the following.

• Construct a ray $\overrightarrow{AR}$ at *an angle*

• In a compass, take *a measurement*

• Using the compass, mark *a number of segments* on the ray.

• Construct line $D$ to $B$

• Construct a parallel to $\overline{DB}$ through the point $C$

• The point of intersection of the parallel through $C$ and $\overline{AB}$ is marked $P$.

The ratio $\overline{AP}:\overline{PB}$ is same as the ratio $\overline{AC}:\overline{CD}$, which is $2:1$

summary

**Segmenting a line in a given ratio (Triangle Method)**: Use the property of similar triangles the given line can be segmented.
Given $\overline{AB}$ to be divided in ratio $m:n$.

The triangle $\u25b3ABD$ is constructed with $\overline{AD}$ having $m+n$ segments. By using the first $m$ segments, the similar triangle $\u25b3ACP$ is created.

Since, $\overline{AC}:\overline{CD}$ is $m:n$, by the similarity property of triangle, $\overline{AP}:\overline{PB}=m:n$

segmenting a line

The objective is to divide the given line $\overline{AB}$ in ratio $m:n$. The properties of similar triangles help in acheiving that.

A set of similar triangles are formed with $\overline{AB}$ and a pair of parallel lines. The formulation is shown in the figure. In such a construction, the following properties are noted.

• $\overline{AD}\mid \mid \overline{BC}$

• $\overline{AP}:\overline{AE}=\overline{PB}:\overline{BF}$

The construction is explained in the following pages.

The objective is to divide the given line $\overline{AB}$ in ratio $\overline{AP}:\overline{PB}=1:2$.

The construction is visualized in the following.

• Construct a ray $\overrightarrow{BC}$ at *an angle*. Any angles will do fine.

• Construct another ray $\overrightarrow{AD}$ in parallel to $\overrightarrow{BC}$.

• In a compass, take *a measurement*. Any arbitrary length will do fine.

• Using the compass, mark *a number of segments* on the rays $\overrightarrow{AD}$ and $\overrightarrow{BC}$ marking points $E$ and $F$.

• Construct line $E$ to $F$

• The point $P$ is marked at the intersection of $\overline{EF}$ and $\overline{AB}$.

The figure illustrates $3$ different angles. The procedure will work for any arbitrary measure of angle.

The construction is visualized as given below.

• Construct a ray $\overrightarrow{BC}$ at an angle

• Construct another ray $\overrightarrow{AD}$ in parallel to the $\overrightarrow{BC}$.

• In a compass, take an arbitrary length on the compass

• Using the compass, mark a number of segments on the rays $\overrightarrow{AD}$ and $\overrightarrow{BC}$ marking points $E$ and $F$.

• Construct line $E$ to $F$

• The point $P$ is marked at the intersection of $\overline{EF}$ and $\overline{AB}$.

The figure illustrates $3$ different measures on compass $\overline{AE1}$, $\overline{AE2}$, $\overline{AE3}$. The procedure to construct will work for any arbitrary length.

Continuing on dividing $\overline{AB}$ in ratio $\overline{AP}:\overline{PB}=1:2$.

• Using the measure on the compass, mark *a number of segments* on the rays $\overrightarrow{AD}$ and $\overrightarrow{BC}$ marking points $E$ and $F$.
To achieve $1:2$ ratio, the points are marked with $1$ segment on $\overrightarrow{AD}$ and $2$ segments on $\overrightarrow{BC}$

From the properties of similar trianges, $\overline{AP}:\overline{PB}=1:2$, same as the ratio $\overline{AE}:\overline{BF}$.

summary

**Segmenting a line in a given Ratio (Traversal of pair of parallels)** : Use the properties of angles in the traversal of parallel lines and the similar triangles, the given line can be segmented. Given $\overline{AB}$ to be divided in ratio $m:n$.

The parallels $\overrightarrow{AD}$ and $\overrightarrow{BC}$ are constructed.

Mark points $E$ and $F$ in the given ratio $m:n$. Since $\overline{AE}:\overline{BF}$ is in the given ratio $m:n$, by the similarity property of triangles, $\overline{AP}:\overline{PB}=m:n$

scale-down a triangle

Consider the given $\u25b3ABC$. The objective is to construct $\u25b3APQ$ such that the corresponding sides are in ratio $3:2$. To achieve that, the first step is to divide the side $\overline{AB}$ to mark point $P$. Then $\overline{PQ}$ can be constructed to complete $\u25b3APQ$.

Considering scaling the given $\u25b3ABC$ to $\u25b3APQ$ in the ratio $3:2$. The given ratio is for the corresponding sides of the triangles, i.e. $\overline{AB}:\overline{AP}$. But to divide the $\overline{AB}$, the ratio is modified to $\overline{AP}:\overline{PB}=2:1$.

Considering scaling the given $\u25b3ABC$ to $\u25b3APQ$ in the ratio $3:2$. The figure illustrates the solution. The $\overline{AB}$ is segmented to $\overline{AP}$ and $\overline{PB}$. This marks the point $P$.

To mark $Q$, construct the line $\overline{PQ}$ in parallel to $\overline{BC}$.

summary

**Scaling a Triangle**:The figure illustrates the solution. The $\overline{AB}$ is segmented to $\overline{AP}$ and $\overline{PB}$. This marks the point $P$.

The $\overline{PQ}$ is constructed in parallel to $\overline{BC}$.

$\u25b3APQ$ is constructed.

scale-up a triangle

Consider the given $\u25b3ABC$. The objective is to construct $\u25b3APQ$ such that the corresponding sides are in the ratio $2:3$. To achieve that, one of the sides can be scaled-up to construct similar triangles.

Scale-up the side $\overline{AB}$ to mark point $P$;. Then $\overline{PQ}$ can be constructed to complete $\u25b3APQ$.

Considering scaling the given $\u25b3ABC$ to $\u25b3APQ$ in the ratio $2:3$. The given ratio is for the corresponding sides of the triangles, i.e. $\overline{AB}:\overline{AP}$. But to scale the $\overline{AB}$, the ratio is modified to $\overline{AB}:\overline{BP}=2:1$.

The figure illustrates the solution. The $\overline{AB}$ is scaled to $\overline{AB}$ and $\overline{BP}$. This marks the point $P$. To mark $Q$, construct the line $\overline{PQ}$ in parallel to $\overline{BC}$ and extend $\overline{AC}$.

The figure illustrates the solution. The $\overline{AB}$ is scaled up to $\overline{AB}$ and $\overline{BP}$. This marks the point $P$.

The $\overline{AC}$ is extended to the ray $\overrightarrow{AT}$. The $\overline{PQ}$ is constructed in parallel to $\overline{BC}$ and that meets $\overrightarrow{AT}$.

$\u25b3APQ$ is constructed.

summary

**Scaling of Triangles** : To scale a triangle, scale one of the sides and use the properties of similar triangles to construct the scaled triangle.

scaling quadrilateral

Consider the given quadrilateral $ABCD$. The objective is to construct the quadrilateral $APRQ$ such that the corresponding sides are in ratio $3:2$. To achieve that, divide the sides, to mark points $P$ and $Q$ and construct parallels to mark point $R$.

Consider the given quadrilateral $ABCD$. The objective is to construct the quadrilateral $APRQ$ such that the corresponding sides are in ratio $3:2$.

The following steps details construction of the quadrilateral $APRQ$.

• Divide the side $AB$ to $2:1$ ratio to mark point $P$.

• Divide the side $AD$ to $2:1$ ratio to mark point $Q$.

• Construct two rays $\overrightarrow{PR}$ and $\overrightarrow{QR}$ in parallel to $\overline{BC}$ and $\overline{DC}$ respectively.

• The point of intersection is point $R$.

Quadrilateral $APRQ$ is constructed.

Consider the given quadrilateral $ABCD$. The objective is to construct the quadrilateral $APRQ$ such that the corresponding sides are in ratio $2:3$. To achieve that scale-up the sides, to mark points $P$ and $Q$ and construct parallels to mark point $R$

Consider the given quadrilateral $ABCD$. The objective is to construct the quadrilateral $APRQ$ such that the corresponding sides are in ratio $2:3$

The following steps details construction of the quadrilateral $APRQ$.

• Scale-up the side $AB$ to $2:1$ ratio to mark point $P$.

• Scale-up the side $AD$ to $2:1$ ratio to mark point $Q$.

• Construct two rays $\overrightarrow{PR}$ and $\overrightarrow{QR}$ in parallel to $\overline{BC}$ and $\overline{DC}$ respectively.

• The point of intersection is point $R$

Quadrilateral $APRQ$ is constructed.

summary

**Scaling of Polygons** : To scale a polygon, scale a side and construct a parallel to the next side and scale the side in the parallel. Continue this with all the sides to complete scaling the given polygon.

scaling a circle

Consider the given circle $A$ centered at point $O$. The objective is to construct circle $B$ such that the radius are in ratio $3:2$. To achieve that, construct a radius and divide it in $2:1$ ratio

Consider the given circle $A$ centered at point $O$. The objective is to construct circle $B$ such that the radius are in ratio $3:2$

The following steps detail scaling a circle.

• Construct a radius $\overline{OP}$.

• Scale the radius in ratio $2:1$ to mark point $Q$.

• Construct a circle with radius $\overline{OQ}$.
Scaled Circle $B$ is constructed.

summary

**Scaling a Circle** : To scale a circle, scale a radius as per scaling a line segment. The scaled circle is constructed with scaled line segment as the radius.

Outline

The outline of material to learn "Consrtruction (High school)" is as follows.

Note: *Click here for detailed overview of "Consrtruction (High school)" *

Note 2: * click here for basics of construction, which is essential to understand this. *

→ __Construction of Triangles With Secondary Information__

→ __Construction: Scaling a line, triangle, polygon, circle__

→ __Construction of Tangents and Chord for a circle__