Mastering Applied Practical Geometry Ex 4.2 (Class 8 Maths)

Understanding Quadrilateral Construction with Diagonals and Sides

In previous classes, you've learned how to construct triangles given various conditions. Building on that knowledge, we now tackle quadrilaterals. A quadrilateral has four sides and four angles. However, unlike triangles, simply knowing four sides isn't enough to define a unique quadrilateral. We often need more information, such as diagonals or angles.

Exercise 4.2 specifically deals with cases where we are provided with two diagonals and three sides of a quadrilateral. The key strategy here is to break down the quadrilateral into simpler, constructible shapes – mainly triangles. Every quadrilateral can be divided into two triangles by drawing one of its diagonals. If we can construct these two triangles accurately, we can then combine them to form the complete quadrilateral.

Always start by drawing a rough sketch of the quadrilateral. This step is absolutely crucial! It helps you visualize the figure, identify which parts are given, and plan your construction steps. Label the vertices and mark the given measurements on your rough sketch. This simple act will save you from confusion and errors during the actual construction. Remember, precision is key in geometry, so use a sharp pencil, a good ruler, and a well-adjusted compass.

Step-by-Step Guide to Construct a Quadrilateral (Given 2 Diagonals and 3 Sides)

1. Step 1: Draw a Rough Sketch — Always begin by drawing a neat, labeled rough sketch of the quadrilateral. Mark all the given side lengths and diagonal lengths clearly on this sketch. This helps in visualizing the construction.
2. Step 2: Construct the First Triangle — Identify one of the triangles within the quadrilateral that can be constructed using the given measurements. Typically, you'll start with a triangle that uses one of the given sides as its base and involves parts of the given diagonals or other sides. For example, if you have sides AB, BC, CD, DA and diagonal AC, you might start with triangle ABC using sides AB, BC and diagonal AC.
3. Step 3: Locate the Third Vertex of the First Triangle — Using your compass, draw arcs from the two known vertices (e.g., A and B) with radii corresponding to the lengths of the remaining two sides of the first triangle (e.g., AC and BC). The intersection of these arcs will give you the third vertex (e.g., C).
4. Step 4: Construct the Second Triangle — Now, consider the second triangle that forms the quadrilateral (e.g., triangle ADC if the first was ABC). Use the diagonal that is common to both triangles (e.g., AC) as one of its sides. You will use the given lengths of the remaining sides and the second diagonal to locate the fourth vertex (e.g., D).
5. Step 5: Locate the Fourth Vertex — From the appropriate vertices (e.g., A and C for locating D), draw arcs with radii equal to the given lengths (e.g., AD and CD). The intersection of these arcs will give you the fourth vertex (e.g., D).
6. Step 6: Complete the Quadrilateral — Join the vertices you've constructed (e.g., A to D, D to C, C to B, B to A). You have successfully constructed the required quadrilateral. Double-check your measurements to ensure accuracy.

Worked Examples: Constructing Quadrilaterals

• Example 1: Construct a quadrilateral LIFT where LI = 4 cm, IF = 3 cm, TL = 2.5 cm, LF = 4.5 cm and IT = 4 cm. Step 1: Rough Sketch Draw a rough sketch of quadrilateral LIFT and label the given measurements: LI = 4 cm, IF = 3 cm, TL = 2.5 cm, LF = 4.5 cm (diagonal), IT = 4 cm (diagonal). Step 2: Construct ΔLIF We have sides LI (4 cm), IF (3 cm) and diagonal LF (4.5 cm). Draw a line segment LI = 4 cm. With L as the center, draw an arc of radius 4.5 cm (LF). With I as the center, draw an arc of radius 3 cm (IF). Let these arcs intersect at F. Join LF and IF. ΔLIF is constructed. Step 3: Locate point T Now we need to locate point T. We know TL = 2.5 cm and IT = 4 cm. With L as the center, draw an arc of radius 2.5 cm (TL). With I as the center, draw an arc of radius 4 cm (IT). Let these arcs intersect at T. Step 4: Complete the Quadrilateral Join LT and FT. LIFT is the required quadrilateral.
• Example 2: Construct a quadrilateral GOLD where OL = 7.5 cm, GL = 6 cm, GD = 6 cm, LD = 5 cm, OD = 10 cm. Step 1: Rough Sketch Draw a rough sketch of quadrilateral GOLD and label the given measurements: OL = 7.5 cm, GL = 6 cm, GD = 6 cm, LD = 5 cm, OD = 10 cm (diagonal). Step 2: Construct ΔGLD We have sides GL (6 cm), LD (5 cm) and GD (6 cm). Draw a line segment GL = 6 cm. With G as the center, draw an arc of radius 6 cm (GD). With L as the center, draw an arc of radius 5 cm (LD). Let these arcs intersect at D. Join GD and LD. ΔGLD is constructed. Step 3: Locate point O Now we need to locate point O. We know OL = 7.5 cm and OD = 10 cm. With L as the center, draw an arc of radius 7.5 cm (OL). With D as the center, draw an arc of radius 10 cm (OD). Let these arcs intersect at O. Step 4: Complete the Quadrilateral Join OG and OL. GOLD is the required quadrilateral.

Exam Tip: Accuracy and Rough Sketches are Your Best Friends!

When constructing quadrilaterals or any geometric figure, two things are absolutely critical for scoring full marks: accuracy and a rough sketch. Always start by drawing a freehand, labeled rough sketch. This helps you visualize the problem, identify which segments form triangles, and plan your construction steps logically. Without a rough sketch, it's very easy to get confused about which arcs to draw from which points. Secondly, use sharp pencils, a well-set compass, and an accurate ruler. Even a millimeter off can make your final figure look incorrect or make it impossible to close the quadrilateral. Practice drawing neat, precise arcs and lines. Check your measurements twice before drawing. Remember, geometry is not just about knowing the steps, but also about executing them perfectly.

Practice Questions with Solutions

• Q: Construct a quadrilateral PQRS where PQ = 5 cm, QR = 4 cm, RS = 4.5 cm, PR = 7 cm and QS = 6 cm. A: Step 1: Draw a rough sketch of PQRS with given measurements. Step 2: Construct ΔPQR. Draw PQ = 5 cm. With P as center, draw arc of 7 cm (PR). With Q as center, draw arc of 4 cm (QR). Intersection is R. Join PR, QR. Step 3: Locate S. We have RS = 4.5 cm and QS = 6 cm. With R as center, draw arc of 4.5 cm. With Q as center, draw arc of 6 cm. Intersection is S. Step 4: Join PS and RS. PQRS is the required quadrilateral. Final answer: Quadrilateral PQRS constructed.
• Q: Construct a quadrilateral ABCD where AB = 6 cm, BC = 4 cm, CD = 6.5 cm, AC = 8 cm and BD = 7 cm. A: Step 1: Draw a rough sketch of ABCD with given measurements. Step 2: Construct ΔABC. Draw AB = 6 cm. With A as center, draw arc of 8 cm (AC). With B as center, draw arc of 4 cm (BC). Intersection is C. Join AC, BC. Step 3: Locate D. We have CD = 6.5 cm and BD = 7 cm. With C as center, draw arc of 6.5 cm. With B as center, draw arc of 7 cm. Intersection is D. Step 4: Join AD and CD. ABCD is the required quadrilateral. Final answer: Quadrilateral ABCD constructed.
• Q: Construct a rhombus BENT where BN = 5.6 cm and ET = 6.5 cm. A: Step 1: Draw a rough sketch of rhombus BENT. A rhombus has all sides equal. Diagonals bisect each other at 90 degrees. Here BN and ET are diagonals. Step 2: Draw ET = 6.5 cm. Since diagonals of a rhombus bisect each other at right angles, find the midpoint M of ET. Draw a perpendicular bisector to ET through M. Step 3: With M as center, open compass to (BN/2) = 5.6/2 = 2.8 cm. Mark points B and N on the perpendicular bisector, 2.8 cm above and below M respectively. Step 4: Join BE, EN, NT, and TB. BENT is the required rhombus. Final answer: Rhombus BENT constructed.
• Q: Construct a quadrilateral MORE where MO = 6 cm, OR = 4.5 cm, ME = 7.5 cm, OE = 7.5 cm and RE = 9 cm. A: Step 1: Draw a rough sketch of MORE with given measurements. Step 2: Construct ΔMOE. Draw MO = 6 cm. With M as center, draw arc of 7.5 cm (ME). With O as center, draw arc of 7.5 cm (OE). Intersection is E. Join ME, OE. Step 3: Locate R. We have OR = 4.5 cm and RE = 9 cm. With O as center, draw arc of 4.5 cm. With E as center, draw arc of 9 cm. Intersection is R. Step 4: Join MR and OR. MORE is the required quadrilateral. Final answer: Quadrilateral MORE constructed.

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