Hard Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Define tangent to a circle.

A tangent to a circle is a line that touches the circle at exactly point and is to the at that point.

Correct! Tangent has exactly one point of contact with the circle.

(2) Which property of tangents is used while constructing tangents from an external point?

The property that tangent is to the at the point of .

Perfect! This creates a right angle, enabling construction using Thales theorem.

(3) State the RHS criterion for constructing a triangle.

RHS construction requires the angle, , and side(s) of a right triangle.

Excellent! RHS = Right angle-Hypotenuse-Side criterion.

(4) Write the steps to construct a 75° angle using only a compass.

Step 1: Construct ° angle.

Step 2: Construct ° angle.

Step 3: Bisect the ° angle between them to get 75°.

Great! 75° = 60° + 15° = 60° + (30°÷2).

(5) What is the locus of points equidistant from two intersecting lines?

The locus is the pair of angle bisectorsan angle bisectoran arc between the intersecting lines of the angles formed by the intersecting lines.

Perfect! Two perpendicular angle bisectors form the complete locus.

Short Answer Questions (2 Marks Each)

(1) Construct a triangle given base 7 cm, base angle 60°, and difference of other two sides 3 cm.

(2) Construct a tangent to a circle of radius 4 cm from a point 6 cm away from the center.

(3) Construct a triangle similar to a given triangle with scale factor 53.

(4) Construct a right triangle whose base is 8 cm and hypotenuse is 10 cm.

(5) Construct an equilateral triangle of side 5 cm and then construct another similar triangle with scale factor 23.

Long Answer Questions (4 Marks Each)

(1) Construct a triangle with sides 6 cm, 7 cm, and 8 cm. Then construct another triangle similar to it with scale factor 43.

(2) Construct a triangle with base 8 cm, base angle 60°, and sum of the other two sides 12 cm.

(3) Construct a right triangle ABC in which AB = 6 cm, AC = 8 cm, and ∠B = 90°. Draw a tangent to the circumcircle of ΔABC from a point 5 cm away from the center.

(4) Construct a triangle given base 5 cm, one adjacent angle 45°, and difference of other two sides 2 cm.

(5) Construct two tangents to a circle of radius 5 cm from an external point 13 cm away from its center. Measure the length of the tangents.

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) In constructing tangents to a circle from an external point, the key property used is:

(a) Radius ⟂ Tangent (b) Diameter bisects chord (c) Perpendicular bisector property (d) Central angle theorem

Correct! Tangent is perpendicular to radius at point of contact.

(2) Constructing a triangle when one side, one adjacent angle, and sum of the other two sides is given belongs to:

(a) Special cases (b) SSS construction (c) SAS construction (d) RHS construction

Correct! Sum/difference problems are special construction cases.

(3) Constructing a tangent from an external point involves drawing:

(a) Right triangle using Pythagoras theorem (b) Isosceles triangle (c) Equilateral triangle (d) Rectangle

Correct! Tangent construction creates right triangle configuration.

(4) The scale factor greater than 1 gives:

(a) Larger similar triangle (b) Smaller similar triangle (c) Same size triangle (d) Impossible construction

Correct! Scale factor > 1 enlarges the triangle.

(5) In RHS construction, which elements are given?

(a) Hypotenuse and one side (b) Two angles (c) Three sides (d) Two sides and an angle

Correct! RHS needs right angle + hypotenuse + one side.

(6) To construct two tangents to a circle from an external point, we use:

(a) Compass and straightedge (b) Protractor only (c) Ruler only (d) Calculator

Correct! All geometric constructions use only compass and straightedge.

(7) The number of tangents that can be drawn to a circle from an external point is:

(a) 2 (b) 3 (c) 4 (d) 1

Correct! Exactly 2 tangents can be drawn from any external point.

(8) To construct a triangle with sides 6 cm, 7 cm, and 8 cm, the method used is:

(a) SSS (b) SAS (c) ASA (d) RHS

Correct! Three sides given → SSS construction method.

(9) The locus of points equidistant from two intersecting lines is:

(a) Angle bisectors (b) Perpendicular bisectors (c) Altitudes (d) Medians

Correct! Angle bisectors are equidistant from the intersecting lines.

(10) The length of each tangent drawn from an external point to a circle of radius 5 cm, when the point is 13 cm from the center, is:

(a) 12 cm (b) 11 cm (c) 10 cm (d) 9 cm

Correct! Tangent length = 132−52 = 169−25 = 12 cm.

Advanced Construction Challenge

Determine whether these statements are True or False:

Advanced Construction Quiz

🎉 Congratulations! What You've Mastered:

You have successfully completed the "Advanced Constructions" hard worksheet and learned:

(1) Advanced Triangle Constructions: Mastering SSS, SAS, ASA, and RHS methods with complex conditions

(2) Circle Tangent Construction: Drawing tangents from external points using Thales theorem and geometric principles

(3) Similar Triangle Construction: Creating triangles with specific scale factors for enlargement and reduction

(4) Special Construction Cases: Handling sum and difference of sides problems with sophisticated techniques

(5) Complex Angle Construction: Creating precise angles like 75° using compass-only methods

(6) Locus Applications: Understanding and applying locus concepts for equidistant points

(7) Pythagorean Applications: Using d2−r2 formula for tangent length calculations

(8) Right Triangle Constructions: Applying RHS criterion with given hypotenuse and side measurements

(9) Geometric Theorem Integration: Connecting constructions to fundamental theorems like Thales and Pythagoras

(10) Scale Factor Mastery: Understanding ratios like 43, 53, 23 for similarity transformations

(11) Perpendicular and Bisector Properties: Advanced applications of perpendicular bisectors and angle bisectors

(12) Circumcircle Constructions: Working with circumcenters and circumradii in right triangles

(13) Mathematical Verification: Calculating and verifying construction results using algebraic methods

(14) Multi-step Problem Solving: Combining multiple construction techniques in complex scenarios

(15) Precision and Accuracy: Achieving exact geometric relationships using only compass and straightedge

Outstanding work! You now have mastery over advanced geometric construction techniques and their mathematical foundations!