Maharashtra Board Class 7 Maths part 1 Chapter 1 Geometrical Constructions PDF Download

Geometrical Constructions Part One

Let's Recall

In previous classes, we have learnt about the line, line segment, angle, angle bisector, etc. We measure an angle in degrees. If angle ABC measures 40 degrees, we write it as m angle ABC = 40 degrees.

Angle Bisector

You see the figure of angle ABC alongside. An angle bisector divides an angle into two equal parts. Is ray BM the bisector of angle ABC?

Teacher's Note

An angle bisector divides an angle into two equal parts. Just like your teacher divides a class into two equal groups, an angle bisector divides an angle equally.

Exam Trick

Remember: Bisector = divide into two equal parts. The word "bi" means two, so bisector means two equal parts.

Points to Remember

An angle bisector divides an angle into two equal parts.
We measure angles in degrees.
A ray BM can be the bisector of angle ABC.
The angle bisector helps us make two equal angles from one angle.

Perpendicular Bisector of a Line Segment

Draw a line segment PS of length 4 cm and draw its perpendicular bisector. Name it line CD.

How will you verify that CD is the perpendicular bisector?

m angle CMS = ___ degrees

Is l(PM) = l(SM)?

Teacher's Note

A perpendicular bisector is a line that cuts another line into two equal parts at 90 degrees. Like when you fold a paper in half, the fold line is the perpendicular bisector.

Exam Trick

Remember: Perpendicular = 90 degrees, Bisector = divide into two equal parts. So perpendicular bisector does both things.

Points to Remember

A perpendicular bisector makes an angle of 90 degrees.
It divides a line segment into two equal parts.
The two parts are equal in length.
Every point on the perpendicular bisector is equidistant from the endpoints.

Let's Learn

The Property of the Angle Bisectors of a Triangle

Activity

1. Draw any triangle PQR.

2. Use a compass to draw the bisectors of all three of its angles. Extend the bisectors, if necessary, so that they intersect one another.

3. These three bisectors pass through the same point. That is, they are concurrent. Name the point of concurrence 'I'. Note that the point of concurrence of the angle bisectors of a triangle is in the interior of the triangle.

4. Draw perpendiculars IA, IB and IC respectively from I on to the sides of the triangle PQ, QR and PR. Measure the lengths of these perpendiculars. Note that IA = IB = IC.

The angle bisectors of a triangle are concurrent. Their point of concurrence is called the incentre, and is shown by the letter 'I'.

The perpendicular bisectors of the sides of a triangle are concurrent. Their point of concurrence is called the circumcentre, and is shown by the letter 'C'.

Teacher's Note

Three angle bisectors of a triangle always meet at one point inside the triangle. This point is called the incentre. It is like the heart of the triangle.

Exam Trick

Remember: Incentre = inside the triangle. The three angle bisectors always meet inside the triangle. This meeting point is the incentre.

Points to Remember

The three angle bisectors of a triangle meet at one point called the incentre.
The incentre is always inside the triangle.
The distance from the incentre to all three sides is equal.
This point is shown by the letter 'I'.

The Property of Perpendicular Bisectors of the Sides of a Triangle

Activity

1. Use a ruler to draw an acute-angled triangle and an obtuse-angled triangle. Draw the perpendicular bisectors of each side of the two triangles.

2. In each triangle, note that the perpendicular bisectors of the sides are concurrent.

3. Name their point of concurrence 'C'. Measure the distance between C and the vertices of the triangle. Note that CX = CY = CZ.

4. Observe the location of the point of concurrence of the perpendicular bisectors.

Teacher's Note

The perpendicular bisectors of the three sides of a triangle always meet at one point. This point is called the circumcentre. It is equidistant from all three vertices.

Exam Trick

Remember: Circumcentre = around the triangle. The circumcentre is the centre of a circle that passes through all three vertices of the triangle.

Points to Remember

The three perpendicular bisectors of a triangle meet at one point called the circumcentre.
The circumcentre is shown by the letter 'C'.
The distance from the circumcentre to all three vertices is equal.
The location of the circumcentre depends on the type of triangle.
In an acute triangle, it is inside. In a right triangle, it is on the hypotenuse. In an obtuse triangle, it is outside.

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