Constructions Class 10 Mathematics Concept notes and Formula Sheet

 

Concepts:

1. Introduction to Constructions:

   • Constructions in geometry involve creating accurate figures and shapes using only a compass, straightedge (ruler), and pencil.
   • Common constructions include drawing perpendiculars, bisectors, and specific angles.

2. Basic Tools for Constructions:

   • Compass: Used to draw arcs and circles.
   • Straightedge (Ruler): Used to draw straight lines. It is not graduated, meaning it does not have measurement markings.
   • Pencil: For marking points, lines, and curves.

3. Basic Constructions:

   • Perpendicular Bisector of a Line Segment:

     • Given a line segment $AB$, you can construct its perpendicular bisector as follows:
       1. With $A$ and $B$ as centers, draw arcs of the same radius greater than half the length of $AB$, intersecting above and below the line segment.
       2. The intersection points of the arcs will form a line that is the perpendicular bisector of $AB$.

   • Angle Bisector:

     • To bisect a given angle $\angle ABC$:
       1. With $B$ as the center, draw an arc that cuts $AB$ and $BC$ at points $P$ and $Q$, respectively.
       2. With $P$ and $Q$ as centers, draw arcs of equal radii that intersect at $R$.
       3. Draw a line from $B$ through $R$. This line bisects $\angle ABC$.

4. Constructing Angles:

   • Constructing a 60° Angle:

     • To construct a 60° angle at a point $A$:
       1. Draw a ray $AB$.
       2. With $A$ as the center, draw an arc that cuts $AB$ at $C$.
       3. Without changing the compass width, draw an arc with $C$ as the center, intersecting the first arc at $D$.
       4. Draw a line from $A$ through $D$ to form $\angle DAB = 60^\circ$.

   • Constructing a 90° Angle:

     • To construct a 90° angle at a point $A$:
       1. Draw a ray $AB$.
       2. With $A$ as the center, draw an arc that cuts $AB$ at $C$.
       3. Without changing the compass width, draw an arc with $C$ as the center, and then draw another arc with the intersection of the first arc as the new center. Repeat until you have four intersections.
       4. Draw a line from $A$ through the fourth intersection to form a 90° angle.

5. Division of a Line Segment:

   • Dividing a Line Segment in a Given Ratio:
     • To divide a line segment $AB$ in the ratio $m:n$:
       1. Draw a ray $AX$ making an acute angle with $AB$.
       2. Mark $m + n$ equal divisions on $AX$ and label them as $A_1, A_2, \ldots, A_{m+n}$.
       3. Join $A_{m+n}$ to $B$.
       4. Through $A_m$, draw a line parallel to $A_{m+n}B$, intersecting $AB$ at $P$.
       5. $P$ divides $AB$ in the ratio $m:n$.

6. Construction of Triangles:

   • Constructing a Triangle Similar to a Given Triangle:

     • To construct a triangle similar to a given triangle $ABC$ with a scale factor $\frac{m}{n}$:
       1. Draw $AB$ and extend it to a new point $D$ such that $\frac{AD}{AB} = \frac{m}{n}$.
       2. Draw an arc with center $D$ and radius equal to the original side $AC$ to intersect $BD$.
       3. Repeat for the other sides to complete the new triangle.

   • Construction of a Triangle Given its Base, Vertical Angle, and Two Sides:

     • Given the base $AB$, vertical angle $\angle C$, and sides $AC$ and $BC$:
       1. Draw the base $AB$.
       2. Draw the angle $\angle C$ at a point on $AB$.
       3. With $A$ and $B$ as centers, draw arcs with radii equal to $AC$ and $BC$.
       4. The intersection of these arcs gives the third vertex of the triangle.

7. Construction of Tangents to a Circle:

   • Tangent to a Circle from a Point Outside the Circle:
     • To draw tangents from a point $P$ outside a circle with center $O$:
       1. Draw the radius $OP$ and find the midpoint $M$ of $OP$.
       2. With $M$ as the center, draw a circle with radius $OM$, intersecting the original circle at points $T_1$ and $T_2$.
       3. Draw lines $PT_1$ and $PT_2$. These are the tangents from $P$ to the circle.

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Formula Sheet:

1. Perpendicular Bisector:

   • To bisect a line segment $AB$: Draw arcs from $A$ and $B$ with a radius greater than half the length of $AB$, intersecting at points. The line through these intersections is the perpendicular bisector.

2. Angle Bisector:

   • To bisect $\angle ABC$: Draw an arc from $B$ intersecting $AB$ and $BC$ and then use the compass to create intersections. The line from $B$ through these intersections is the angle bisector.

3. Constructing Specific Angles:

   • 60° Angle: Draw arcs from a point on a ray and intersect these arcs to form the 60° angle.
   • 90° Angle: Use a similar process with additional arcs to create a 90° angle.

4. Division of a Line Segment:

   • To divide $AB$ in $m:n$: Use a ray and equal segments, connecting the last point to $B$ and drawing a parallel through the desired segment.

5. Constructing Similar Triangles:

   • To construct a triangle similar to a given triangle with a specific ratio: Extend the base and use arcs to form the new triangle according to the ratio.

6. Tangents from a Point to a Circle:

   • To draw tangents from $P$ to a circle: Use the midpoint of $OP$ to create a circle intersecting the original one, then draw lines from $P$ to the intersection points.