Practical Geometry: Constructing Quadrilaterals (Ex 4.4)

Understanding the Core Concept: Three Sides and Two Included Angles

To construct a unique quadrilateral, we generally need five measurements. In this specific case (NCERT Exercise 4.4), those five measurements are three sides and two included angles. What does 'included angle' mean? An included angle is the angle formed at the vertex where two sides meet. For example, if you have sides AB and BC, the included angle is ∠B. The condition for this construction is that we must know the lengths of three sides that are in a sequence (one after another) and the two angles that are 'sandwiched' between them. For instance, to construct quadrilateral ABCD, we would need to know the lengths of AB, BC, CD, and the angles ∠B and ∠C. This specific set of information locks the shape into place, preventing it from changing. The key is to start with the side that connects the two known angles.

How to Construct a Quadrilateral (3 Sides, 2 Included Angles)

1. Step 1: Draw a Rough Sketch — Before touching your geometry tools, always draw a rough, freehand sketch of the quadrilateral. Label it with the given measurements (sides and angles). This will act as your guide and help you visualise the final figure.
2. Step 2: Draw the Base Side — Identify the side that lies between the two given angles. Draw this line segment first using a ruler. For example, if you are given sides AB, BC, CD and angles ∠B and ∠C, you should start by drawing the line segment BC.
3. Step 3: Construct the First Angle and Side — At one endpoint of the base (say, B), use a protractor and compass to construct the first given angle (∠B). Then, along this new ray, measure and cut an arc equal to the length of the adjacent side (AB). Mark this point (A).
4. Step 4: Construct the Second Angle and Side — Now, go to the other endpoint of the base (C). Construct the second given angle (∠C). Along this new ray, measure and cut an arc equal to the length of the other adjacent side (CD). Mark this point (D).
5. Step 5: Complete the Quadrilateral — You now have three sides and four vertices (A, B, C, and D). Simply join the last two points (A and D) with a ruler to form the fourth side. You have successfully constructed the required quadrilateral!

Worked Example: Constructing Quadrilateral DEAR

• Let's construct Quadrilateral DEAR with DE = 4 cm, EA = 5 cm, AR = 4.5 cm, ∠E = 60°, and ∠A = 90°. Step 1: Rough Sketch First, draw a rough sketch of the quadrilateral and label the given sides and angles. Notice that the side EA is between the two given angles ∠E and ∠A. Step 2: Draw the Base Using a ruler, draw a line segment EA of length 5 cm. Step 3: Construct ∠E and side DE Place the protractor at point E and draw a ray EX making an angle of 60° with EA. With E as the center and a radius of 4 cm (length of DE), draw an arc that cuts the ray EX. Mark this intersection point as D. Step 4: Construct ∠A and side AR Place the protractor at point A and draw a ray AY making an angle of 90° with EA. With A as the center and a radius of 4.5 cm (length of AR), draw an arc that cuts the ray AY. Mark this intersection point as R. Step 5: Join the final points Join D and R using a ruler. Final Figure: DEAR is the required quadrilateral constructed with the given measurements.

Tips for Acing Construction Questions

In the exam, marks are often given for each correct step of construction. Always write down the steps of construction as you perform them. Make sure your construction lines (the arcs and rays) are visible but light, and the final quadrilateral is drawn darker. Use a sharp pencil for accuracy. If you are asked to construct an angle like 60°, 90°, or 120°, it's better to use a compass and ruler rather than a protractor, as it demonstrates a better understanding of geometric principles.

Practice Questions with Solutions

• Q: Construct quadrilateral ABCD where AB = 4.5 cm, BC = 5.2 cm, CD = 5 cm, ∠B = 105°, and ∠C = 80°. A: Step 1: Draw a rough sketch and label the sides and angles. Identify BC as the base side between the two angles. Step 2: Draw a line segment BC = 5.2 cm. Step 3: At point B, construct an angle of 105°. Measure and cut a line segment BA = 4.5 cm on the ray. Step 4: At point C, construct an angle of 80°. Measure and cut a line segment CD = 5 cm on this new ray. Step 5: Join point A to point D. Final answer: ABCD is the required quadrilateral.
• Q: Construct quadrilateral PQRS where QR = 6 cm, PQ = 4 cm, RS = 5 cm, ∠Q = 45°, and ∠R = 90°. A: Step 1: Draw a rough sketch. The side between the angles is QR. Step 2: Draw line segment QR = 6 cm. Step 3: At Q, construct an angle of 45°. From Q, mark point P on the ray such that PQ = 4 cm. Step 4: At R, construct an angle of 90°. From R, mark point S on the ray such that RS = 5 cm. Step 5: Join P and S with a straight line. Final answer: PQRS is the constructed quadrilateral.
• Q: Can you construct a quadrilateral LION where LI = 4 cm, IO = 5 cm, ON = 4.5 cm, ∠I = 75°, and ∠O = 185°? Explain your reasoning. A: Step 1: Analyze the given measurements. We have three sides (LI, IO, ON) and two included angles (∠I and ∠O). Step 2: Check the angles. An angle in a convex quadrilateral must be less than 180°. Here, ∠O is given as 185°, which is a reflex angle. Step 3: Conclude based on the properties of a standard quadrilateral. While a shape can have a reflex angle, standard construction methods taught in this chapter are for convex quadrilaterals. Final answer: No, we cannot construct a standard convex quadrilateral because one of the interior angles (∠O = 185°) is greater than 180°. Construction would result in a non-convex (or re-entrant) quadrilateral, which is generally not what is intended in these exercises.
• Q: Construct a quadrilateral TRUE with TR = 3.5 cm, RU = 3 cm, UE = 4 cm, ∠R = 75°, ∠U = 120°. A: Step 1: Draw a rough sketch and label it. RU is the side between the two given angles. Step 2: Draw a line segment RU = 3 cm. Step 3: At point R, construct an angle of 75°. Measure and cut a line segment RT = 3.5 cm on the ray. Step 4: At point U, construct an angle of 120°. Measure and cut a line segment UE = 4 cm on this new ray. Step 5: Join point T to point E. Final answer: TRUE is the required quadrilateral.

Frequently Asked Questions