Applied Practical Geometry Class 8 NCERT

Why Do We Need Five Measurements?

In earlier classes, you learned that three measurements (like SSS, SAS, or ASA) are sufficient to construct a unique triangle. However, for a four-sided figure like a quadrilateral, four measurements are not enough. If you link four rigid sticks of lengths 3 cm, 4 cm, 5 cm, and 6 cm, you can still wiggle the joints to create infinitely many different shapes with different angles. To fix or 'lock' the shape completely, you need a fifth measurement—such as a diagonal or an angle. In applied practical geometry class 8 ncert, you will construct quadrilaterals under five distinct cases: 1) when four sides and one diagonal are given, 2) when two diagonals and three sides are given, 3) when two adjacent sides and three angles are given, 4) when three sides and two included angles are given, and 5) special cases like a square or rhombus with fewer given parameters due to their inherent symmetry.

Step-by-Step Construction Process (Four Sides & One Diagonal)

1. Step 1: Draw a Rough Sketch — Always draw a quick freehand sketch of the quadrilateral first and label all the given measurements. This helps you visualize which triangles can be constructed first using SSS or SAS criteria.
2. Step 2: Construct the Base Triangle — Identify a triangle within the quadrilateral where all three side lengths (or two sides and an angle) are known. Draw the base segment using a ruler, and use a compass to mark arcs for the third vertex.
3. Step 3: Locate the Fourth Vertex — From the endpoints of the constructed triangle, use the remaining two measurements to draw intersecting arcs. The point of intersection is your fourth vertex.
4. Step 4: Join the Vertices — Use a ruler to join the new vertex to the adjacent vertices to complete the quadrilateral. Double-check all lengths with a ruler.

Worked Construction Examples

• Example 1: Construct a quadrilateral ABCD where AB = 4.5 cm, BC = 5.5 cm, CD = 4 cm, AD = 6 cm, and diagonal AC = 7 cm. Step 1: Draw a rough sketch of ABCD and mark the measurements. Notice that AC divides ABCD into two triangles: ABC and ADC. Step 2: First, construct triangle ABC. Draw base line segment AC = 7 cm. Step 3: With A as center and radius 4.5 cm (length AB), draw an arc above AC. Step 4: With C as center and radius 5.5 cm (length BC), draw another arc intersecting the previous arc at point B. Join AB and BC. Step 5: Now, construct triangle ADC. With A as center and radius 6 cm (length AD), draw an arc below AC. Step 6: With C as center and radius 4 cm (length CD), draw an arc intersecting the previous arc at point D. Join AD and CD. Step 7: ABCD is the required unique quadrilateral.

Important Construction Rules & Exam Tips

1. The Triangle Inequality Rule: Before you begin constructing, check if the sum of any two sides of the internal triangles is strictly greater than the third side (especially when diagonals are involved). If this condition fails, construction is mathematically impossible!

2. Keep Your Pencil Sharp: A blunt pencil leads to thick lines and errors of 1–2 mm, which can lose you marks in your CBSE practical geometry exam.

3. Do Not Rub Out Arc Marks: Examiners look for construction arcs to verify that you used a compass and didn't just draw freehand. Keep your arc marks visible and clean.

Practice Questions with Solutions

• Q: Construct a quadrilateral PQRS where PQ = 4 cm, QR = 6 cm, RS = 5 cm, PS = 5.5 cm and PR = 7 cm. A: Step 1: Analyze the given dimensions. We have four sides and one diagonal (PR = 7 cm). We can divide this into triangles PQR and PSR. Step 2: Draw a line segment PR = 7 cm. Step 3: Construct triangle PQR. With P as center, draw an arc of radius 4 cm. With R as center, draw an arc of radius 6 cm. Mark their intersection point as Q. Join PQ and RQ. Step 4: Construct triangle PSR. With P as center, draw an arc of radius 5.5 cm on the opposite side of PR. With R as center, draw an arc of radius 5 cm. Mark their intersection as S. Join PS and RS. Final answer: PQRS is the constructed quadrilateral.
• Q: Can you construct a quadrilateral ABCD with AB = 3 cm, BC = 4 cm, CD = 5.5 cm, DA = 6 cm, and diagonal AC = 10 cm? Justify your answer. A: Step 1: Let us check if triangle ABC can exist with sides AB = 3 cm, BC = 4 cm, and AC = 10 cm. Step 2: Apply the triangle inequality theorem: The sum of any two sides of a triangle must be greater than the third side. Step 3: Test AB + BC: 3 cm + 4 cm = 7 cm. Here, 7 cm is less than AC (10 cm). Step 4: Since the sum of two sides (7 cm) is not greater than the third side (10 cm), triangle ABC cannot be formed. Final answer: No, the construction of quadrilateral ABCD is not possible because the triangle inequality condition is violated for triangle ABC.
• Q: Construct a rhombus BEND where diagonals BN = 5.6 cm and DE = 6.5 cm. A: Step 1: Recall the special property of a rhombus: Its diagonals bisect each other at right angles (90 degrees). Step 2: Draw the diagonal DE = 6.5 cm. Step 3: Construct the perpendicular bisector of DE. To do this, take a compass, open it to more than half of DE, and draw arcs above and below DE from both endpoints D and E. Join the intersection points of the arcs. Let this bisector line intersect DE at point O. Step 4: The other diagonal BN is 5.6 cm. Since diagonals bisect each other, OB = ON = 5.6 / 2 = 2.8 cm. Step 5: With O as center and a radius of 2.8 cm, draw arcs on both sides of the perpendicular bisector to mark points B and N. Step 6: Join BD, BE, ND, and NE. Final answer: BEND is the required constructed rhombus.
• Q: Construct a parallelogram HEAR where HE = 5 cm, EA = 6 cm, and angle R = 85 degrees. A: Step 1: Use the properties of a parallelogram: Opposite sides are equal (HE = AR = 5 cm, EA = HR = 6 cm), and opposite angles are equal (angle E = angle R = 85 degrees). Adjacent angles are supplementary (angle H = 180 - 85 = 95 degrees). Step 2: Draw the base HE = 5 cm. Step 3: At point E, construct an angle of 85 degrees using a protractor and draw a ray EX. Step 4: Cut an arc of radius 6 cm (length EA) along ray EX from E. Mark this point as A. Step 5: From point A, draw an arc of radius 5 cm (as opposite side AR = 5 cm). From point H, draw an arc of radius 6 cm (as opposite side HR = 6 cm). Mark the intersection of these arcs as R. Step 6: Join AR and HR. Final answer: HEAR is the required constructed parallelogram.

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