CBSE Class 9 Mathematics Constructions Notes Set 09

Why Constructions Are Done with Ruler and Compasses

In geometry a construction is a drawing made with only an unmarked ruler and a pair of compasses. The protractor is used only for checking. Every step is justified by a geometric theorem.

Bisector of an Angle

Angle of 60° and Other Angles

1. Draw a ray AB. With A as centre and any radius, draw an arc cutting AB at P.
2. With P as centre and the same radius, cut the arc at Q. Then ∠QAP = 60°, because triangle APQ is equilateral.
3. Bisecting 60° gives 30°, and bisecting 30° gives 15°.
4. For 90°, mark Q and then R by two equal steps on the arc (60° and 120°) and bisect the angle QAR; the bisector is at 90° from AB.
5. Angles such as 105° (90° + 15°) and 67.5° (half of 135°) are built by adding or bisecting these basic angles.

1. With A and B as centres and a radius greater than half of AB, draw arcs on both sides of AB.
2. Mark the points P and Q where the arcs intersect.
3. Join PQ. It meets AB at R, which is the mid-point of AB, and PQ is perpendicular to AB.

Angle of 60° and Other Angles

1. Draw a ray AB. With A as centre and any radius, draw an arc cutting AB at P.
2. With P as centre and the same radius, cut the arc at Q. Then angle QAP = 60° because triangle APQ is equilateral.
3. Bisecting 60° gives 30° and bisecting 30° gives 15°. A 90° angle is made by bisecting the angle between a 60° arc and its neighbouring 60° arc, which is 120°, to get 60° and then adding... in practice by marking a second 60° arc from Q and bisecting the angle between them; 120° is obtained by marking two arcs of 60° each; 105° = 90° + 15°, and 67.5° is half of 135° (90° + 45°).

Constructing Triangles

A triangle can be drawn when suitable three of its six parts (3 sides and 3 angles) are given. Three angles alone are not enough, because infinitely many similar triangles have the same angles.
The base, a base angle and the sum of the other two sides: draw the base, make the angle at one end, cut off the given sum along the ray, join the far end to the other end of the base and draw the perpendicular bisector of that line. Its meeting point with the ray is the third vertex.
The base, a base angle and the difference of the other two sides: draw the base and the angle, cut off the difference along the ray (on the side of the base angle, or on its extension), join the point to the other end of the base and take the perpendicular bisector. Its intersection with the ray is the third vertex.
The perimeter and the two base angles: draw a segment equal to the perimeter, make the given angles at its two ends, bisect them and let the bisectors meet at P. The perpendicular bisectors of AP and BP meet the segment at Q and R, and PQR is the triangle.

Conditions for Existence

A triangle exists only if the sum of any two sides exceeds the third side. A right triangle exists only if the hypotenuse is the longest side.
A right triangle with given base and given sum of hypotenuse and other side is possible only if the sum is greater than the base.