NCERT Solutions Class 7 Maths Chapter 10 Practical Geometry Exercise 10.2

Welcome to your guide for Practical Geometry Exercise 10.2 of Class 7 CBSE Maths! In this topic, we will explore the Side-Side-Side (SSS) construction criterion. This criterion allows us to construct a unique triangle when the lengths of all three sides are known. Learning to construct shapes accurately is a fundamental skill in mathematics that improves spatial reasoning and hand-eye coordination. With YoLearn AI, we will break down each step using a ruler and compass. By the end of this page, you will master the precise steps to construct these triangles, know how to check if a triangle is possible using the Triangle Inequality Theorem, and gain the confidence to ace your exams. Let's open our geometry boxes and begin!

Understanding the SSS (Side-Side-Side) Triangle Construction

To construct a unique triangle, we need specific measurements. The Side-Side-Side (SSS) criterion states that if the lengths of all three sides of a triangle are given, we can construct the triangle uniquely. But why do we need a compass alongside a ruler? If you try to draw a triangle with sides of 4 cm, 5 cm, and 6 cm using only a ruler, you will find it extremely difficult to get the sides to meet at the exact right angle.

Our compass solves this problem by tracing all possible points at a fixed distance from a starting point. Before you begin drawing, always verify if the triangle can actually exist using the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If this condition is not met (for example, if sides are 2 cm, 3 cm, and 6 cm), the arcs you draw with your compass will never intersect, and no triangle can be formed!

Step-by-Step Method for SSS Triangle Construction

  1. Draw a Rough Sketch — Always start by drawing a quick freehand sketch of the triangle. Label the vertices (such as A, B, and C) and write down the given measurements. This acts as a visual map for your final drawing.
  2. Draw the Base Line Segment — Choose one of the given sides (usually the longest side for stability) to be the base. Use your ruler to draw a line segment of this length. Label the endpoints clearly (e.g., B and C) and write its measurement.
  3. Swing the First Arc — To locate the third vertex (A), adjust your compass width to equal the length of the second side (e.g., AB). Place the metal pointer of your compass on point B and draw a wide arc above the base line.
  4. Swing the Second Arc to Intersect — Now, adjust your compass width to match the length of the third side (e.g., AC). Place the metal pointer on point C and draw another arc. Make sure this arc crosses the first arc. Label the intersection point as A.
  5. Connect the Points — Use your ruler to draw neat straight lines from point A to point B, and from point A to point C. Double-check your measurements with a ruler. Your triangle ABC is now successfully constructed!

Important Tips to Avoid Common Construction Mistakes

  1. Keep Your Pencil Sharp: A blunt pencil can make your measurements off by 1 to 2 millimeters, which can lead to losing marks in exams. Use a well-sharpened, thin-lead pencil in your compass.
  2. Check Triangle Inequality First: If an exam question asks you to construct a triangle with sides 2 cm, 3 cm, and 6 cm, do not start drawing. First check: 2 + 3 = 5 cm, which is less than 6 cm. Write down that the construction is impossible because the sum of any two sides must be greater than the third side.
  3. Do Not Erase Your Construction Arcs: Teachers look for clean, clear intersection arcs to verify that you used a compass and did not just guess the lines. Keep your arcs visible, but draw them lightly.

Practice Questions with Solutions

  • Q: Construct ΔXYZ in which XY = 4.5 cm, YZ = 5 cm, and ZX = 6 cm. A: Step 1: Draw a rough sketch of ΔXYZ and label the measurements. Step 2: Draw the base line segment YZ = 5 cm using a ruler. Step 3: Set your compass width to 4.5 cm (the length of XY). Place the metal pointer at Y and draw an arc above the base YZ. Step 4: Set your compass width to 6 cm (the length of ZX). Place the metal pointer at Z and draw an arc that cuts the first arc at point X. Step 5: Use a ruler to join XY and XZ. Final answer: ΔXYZ is the required triangle.
  • Q: Construct an equilateral triangle of side 5.5 cm. A: Step 1: An equilateral triangle has three equal sides. So, the three side lengths are 5.5 cm, 5.5 cm, and 5.5 cm. Let's call it ΔABC. Step 2: Draw the base line segment BC = 5.5 cm using a ruler. Step 3: Open your compass to a width of 5.5 cm. Place the pointer on B and draw an arc above BC. Step 4: Keeping the same compass width of 5.5 cm, place the pointer on C and draw another arc cutting the first arc at point A. Step 5: Join AB and AC using a ruler. Final answer: ΔABC is the required equilateral triangle.
  • Q: Draw ΔPQR with PQ = 4 cm, QR = 3.5 cm, and PR = 4 cm. What type of triangle is this? A: Step 1: Analyze the sides. Since PQ = PR = 4 cm, two sides are equal. This is an isosceles triangle. Step 2: Draw the base line segment QR = 3.5 cm using a ruler. Step 3: Adjust the compass to 4 cm. Place the pointer on Q and draw an arc above QR. Step 4: Keep the compass width at 4 cm, place the pointer on R, and draw an arc intersecting the first arc at point P. Step 5: Join PQ and PR using a ruler. Final answer: ΔPQR is the constructed triangle. Since PQ = PR, it is an isosceles triangle.
  • Q: Can you construct a triangle ABC with sides AB = 3 cm, BC = 2 cm, and AC = 6 cm? Justify your answer. A: Step 1: Apply the Triangle Inequality Property. Check if the sum of the two smaller sides is greater than the largest side. Step 2: The two smaller sides are AB = 3 cm and BC = 2 cm. Sum = 3 cm + 2 cm = 5 cm. Step 3: Compare this sum with the third side AC = 6 cm. Since 5 cm < 6 cm, the sum is smaller than the third side. Final answer: No, triangle ABC cannot be constructed because the sum of the lengths of any two sides of a triangle must always be greater than the length of the third side.

Frequently Asked Questions

What is the SSS criterion in practical geometry?

The SSS (Side-Side-Side) criterion states that a triangle can be uniquely constructed if the lengths of all three sides are known. Using a ruler and compass, we draw arcs based on these lengths to locate the three vertices.

Why do we use a compass instead of just a ruler for drawing triangles?

A ruler can measure flat distances, but it cannot easily find where two lines meet at the correct angles. A compass lets us draw arcs that represent all possible points at a set distance, allowing us to find the exact intersection point where the vertices meet.

What should I do if my compass arcs do not intersect?

If your arcs do not intersect, double-check your measurements first. If your measurements are correct, check the Triangle Inequality Theorem; if the sum of the two shorter sides is equal to or less than the longest side, a triangle cannot be formed.