Math's: Division of a Line Segment: In a Ratio and Construction of Triangles

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Division of a Line Segment in the Given Ratio Internally

To divide a line segment in a given ratio say , follow the steps given below-

  • Step 1: Draw a ray making an acute angle with .

  • Step 2: Starting with , mark off points and on at equal distances from the point .

  • Step 3: Join and .

  • Step 4: Through (i.e. the second point), draw parallel to meeting at .

Division a Line Segment

Division a Line Segment

The point divides the line in the ratio

Construction of Triangles

SSS Case (when three sides are given)- To construct a in which , and , follow the given steps

Construction of Triangles

Construction of Triangles

  • Step 1: Take any of the sides as base. Here, is taken as a base. Draw .

  • Step 2: With as centre and radius , cut an arc.

  • Step 3: With as centre and radius cut another arc intersecting the arc of Step 2 at .

  • Step 4: Join and .

Then, is the required triangle.

SAS Case (when two sides and the included angle is given)- To construct a triangle in which , and

Construction of Triangles

Construction of Triangles

  • Step 1: Take one of the sides as base (e.g. take . Draw

  • Step 2: At , construct an angle

  • Step 3: With as centre and radius draw an arc cutting at .

  • Step 4: Join

Then, is the required triangle.

ASA Case (when two angles and the included side is given)- To construct a triangle when two angles and one side are say , and , go through the following steps

Construction of Triangles

Construction of Triangles

  • Step 1: Draw .

  • Step 2: At , construct

  • Step 3: At , construct meeting at .

Then, ABC is the required triangle.

Right Triangle Case (when the hypotenuse and a side is given)- To construct a right triangle , right angled at , side and hypotenuse , follow the steps given below

Construction of Triangles

Construction of Triangles

  • Step 1: Draw

  • Step 2: At , construct

  • Step 3: With as centre and radius cut an arc intersecting at .

  • Step 4: Join

is the required triangle.

When two sides and a median corresponding to one of these sides, are given- To construct a in which , and median , follow the given steps

Construction of Triangles

Construction of Triangles

  • Step 1: Draw

  • Step 2: Draw the perpendicular bisector of meeting AB in D.

  • Step 3: With as centre and radius cut an arc.

  • Step 4: With as centre and radius cut another arc intersecting the arc of Step 3 at

  • Step 5: Join AC and BC.

Then, is required triangle.

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