NCERT Solutions Class 9 Maths Chapter 11 — Constructions

Solution:

Step 1: An angle bisector is a ray that divides an angle into two equal parts.

Step 2: A key property of an angle bisector is that any point lying on it is equidistant from the two arms of the angle.

Step 3: This property is often used in proofs and constructions involving angle bisectors.

Solution:

Step 1: To construct a perpendicular bisector, arcs must be drawn from both endpoints (X and Y) with the same radius.

Step 2: This ensures that the intersection points of the arcs are equidistant from X and Y, which is a property of points on the perpendicular bisector.

Step 3: Using different radii would lead to intersection points that are not necessarily equidistant, resulting in a line that is not the perpendicular bisector.

Solution:

Step 1: Classical Euclidean constructions are strictly limited to using only two tools: an unmarked ruler (straightedge) and a compass.

Step 2: The unmarked ruler is used to draw straight lines, and the compass is used to draw circles or arcs and transfer distances.

Step 3: Marked rulers (for measuring lengths) and protractors (for measuring angles) are not allowed in these classical constructions.

Solution:

Step 1: The standard construction for a 60° angle involves creating an equilateral triangle.

Step 2: Step 1: With O as center and any convenient radius, draw an arc cutting OA at P. (Correct)

Step 3: Step 2: With P as center and the *same* radius (not a different one), draw an arc cutting the previously drawn arc at Q. This forms an equilateral triangle OPQ where OP=PQ=QO.

Step 4: Step 3: Draw ray OQ. (Correct)

Step 5: Step 4: Angle AOQ will be 60°. (Correct)

Step 6: The error lies in option B, which states 'a radius equal to the first one', implying that the step itself is correct. The question asks for the *incorrect* statement. The statement 'With P as center and a radius *different* from the first one, draw an arc cutting the previously drawn arc at Q.' would be the incorrect step. Let's rephrase the options to correctly identify the incorrect step. The provided options are correct statements of the steps except for one that describes a wrong action. Given the options, the statement that *describes* the correct action (radius equal) is not the incorrect step. Let's assume the question meant to ask which of these *describes an incorrect action*.

Step 7: Revisiting the question: 'which of the following steps is *incorrect*?'. The options describe steps. If option B describes using the 'same radius', then it's a correct step. If it describes using a 'different radius', then it's an incorrect step. Let's assume the provided options have an intended error for one to be picked. The standard construction requires the *same* radius. If an option *states* 'different radius', that's the incorrect step. The current option B states 'radius equal to the first one', which is correct. I need to ensure an option clearly states the *incorrect* action.

Step 8: Let me adjust the options for Q4 to make one clearly describe an incorrect action, as per the initial intention.

Step 9: Revised Q4 options for correct answer D (as per my distribution plan). The original description of option B in my thoughts was 'radius *different* from the first one'. The problem text has 'radius equal'. I must fix this.

Step 10: Corrected option D will describe the incorrect step: 'With P as center and a radius *different* from the first one, draw an arc cutting the previously drawn arc at Q.'

Step 11: The correct answer to this question, if the options were: A. With O as center and any convenient radius, draw an arc cutting OA at P. B. Draw a ray OQ. C. The angle AOQ formed will be 60°. D. With P as center and a radius *different* from the first one, draw an arc cutting the previously drawn arc at Q. Then D would be the incorrect step. My JSON option B states 'radius equal', which is a correct step. I need to make sure one option describes an *incorrect action*.

Step 12: Let's re-evaluate question 4. The initial internal thought for Q4 was 'With P as center and a radius *different* from the first one, draw an arc cutting the previously drawn arc at Q.' making B the error. If the option given is 'equal to the first one', it's a correct step. I need one option to be an incorrect step. Let's change one option to reflect an incorrect action.

Step 13: Option D in the JSON will be the incorrect step description. This aligns with the chosen distribution D.

Solution:

Step 1: An angle of 45° is half of 90°.

Step 2: Therefore, to construct a 45° angle, one must first accurately construct a 90° angle.

Step 3: After constructing the 90° angle, its bisector can be drawn to obtain two 45° angles.