CBSE Class 10 Maths: Constructions Ex 11.1 – Dividing a Line Segment

Understanding Line Segment Division

Dividing a line segment in a given ratio means finding a point on that segment that splits it into two smaller segments, whose lengths are in a specific proportion. For example, if you divide a line segment AB in the ratio 3:2, you're looking for a point P on AB such that AP:PB = 3:2. This concept is fundamental because it connects directly to ideas of similarity and proportion, which are cornerstones of geometry.

The method we use for this construction relies heavily on the Basic Proportionality Theorem (BPT), also known as Thales Theorem, or concepts of similar triangles. Essentially, we create a setup where parallel lines cut transversal lines proportionally, thereby achieving our desired ratio. This elegant approach allows us to perform precise division without relying on measuring scales, upholding the pure spirit of classical geometric constructions.

Step-by-Step Construction: Dividing a Line Segment in a Given Ratio (m:n)

1. Step 1: Draw the Line Segment — Draw any line segment AB of the given length. For example, if you need to divide a 7.6 cm segment, draw AB = 7.6 cm.
2. Step 2: Draw an Acute Ray — Draw a ray AX starting from point A, making an acute angle with line segment AB. The angle size doesn't matter, but it should be acute (less than 90 degrees).
3. Step 3: Mark Points on the Ray — Let the given ratio be m:n. You need to mark (m + n) points on the ray AX at equal distances. Using a compass, starting from A, mark points A₁, A₂, A₃, ..., A_(m+n) such that AA₁ = A₁A₂ = A₂A₃ = ... and so on. The exact length of these arcs doesn't matter, just that they are all equal.
4. Step 4: Join the Last Point to B — Join the point B to the last marked point on the ray AX, which is A_(m+n). So, draw a line segment from B to A_(m+n).
5. Step 5: Draw a Parallel Line — Now, draw a line through the point A_m (the m-th point from A on ray AX) that is parallel to BA_(m+n). To do this, use a compass to construct an angle equal to ∠AA_(m+n)B at A_m, ensuring the new line is parallel. Let this parallel line intersect AB at point P.
6. Step 6: Verify the Division — The point P divides the line segment AB in the desired ratio m:n. That is, AP/PB = m/n. You can verify this using the Basic Proportionality Theorem (BPT) or the concept of similar triangles.

The Mathematical Justification: Why This Construction Works

The elegance of this construction lies in its foundation on the Basic Proportionality Theorem (BPT), also known as Thales Theorem, or simply the property of similar triangles. Let's briefly understand the logic:

When we draw ray AX and mark points A₁, A₂, ..., A_(m+n) at equal intervals, we are setting up a proportionality along the ray. By joining B to A_(m+n), we form a triangle ΔABA_(m+n). Then, we draw a line through A_m parallel to BA_(m+n), intersecting AB at P.

Consider the triangle ΔAA_(m+n)B. By construction, the line segment PA_m is parallel to BA_(m+n). According to the Basic Proportionality Theorem, if a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides proportionally.

In ΔABA_(m+n), since PA_m || BA_(m+n), we have:
AP / PB = AA_m / A_m A_(m+n)

Since we marked the points A₁, A₂, ... at equal distances, AA_m represents 'm' equal parts, and A_m A_(m+n) represents 'n' equal parts. Therefore, AA_m / A_m A_(m+n) = m/n.

Substituting this back, we get AP / PB = m/n. This beautifully confirms that point P divides the line segment AB in the ratio m:n, exactly as intended. This mathematical grounding ensures the accuracy and validity of our geometric construction.

YoLearn AI Tutor Exam Tip for Constructions

Precision is paramount in construction problems! Always use a sharp pencil and a good quality, well-maintained compass and ruler. Dull pencils lead to thick lines, making accurate intersections difficult to identify. Do not erase construction lines; they are part of your working and show your method. Make sure your arcs are clear and your parallel lines are truly parallel. Practice drawing angles of various sizes without a protractor to get comfortable with compass and ruler techniques. When writing down steps, be clear and concise, following the exact sequence of your drawing. Always double-check if your final constructed figure meets all the given conditions.

Practice Questions with Solutions

• Q: Draw a line segment of length 7.6 cm and divide it in the ratio 5:8. Measure the two parts. A: Step 1: Draw a line segment AB = 7.6 cm. Step 2: Draw a ray AX making an acute angle with AB. Step 3: Mark (5+8) = 13 points A₁, A₂, ..., A₁₃ on AX such that AA₁ = A₁A₂ = ... = A₁₂A₁₃. Step 4: Join B to A₁₃. Step 5: Through A₅, draw a line parallel to BA₁₃ intersecting AB at P. (To draw a parallel line, construct ∠AA₅P = ∠AA₁₃B). Step 6: P is the required point. Measure AP and PB. You should find AP ≈ 2.9 cm and PB ≈ 4.7 cm (approximate due to drawing variations, but ratio 5:8). Final answer: The line segment is divided into two parts of approximately 2.9 cm and 4.7 cm.
• Q: Construct a line segment of length 10 cm and divide it in the ratio 3:2. Write down the steps of construction. A: Step 1: Draw a line segment AB = 10 cm. Step 2: Draw a ray AX making an acute angle with AB. Step 3: Mark (3+2) = 5 points A₁, A₂, A₃, A₄, A₅ on AX such that AA₁ = A₁A₂ = A₂A₃ = A₃A₄ = A₄A₅. Step 4: Join B to A₅. Step 5: Through A₃, draw a line parallel to BA₅, intersecting AB at P. (By making ∠AA₃P = ∠AA₅B). Step 6: Point P divides AB in the ratio 3:2. Final answer: The line segment AB is divided at P such that AP:PB = 3:2.
• Q: Divide a line segment of length 8 cm in the ratio 4:3. Justify your construction. A: Step 1: Draw AB = 8 cm. Draw ray AX. Mark 7 points A₁, ..., A₇ on AX at equal distances. Step 2: Join BA₇. Draw a line through A₄ parallel to BA₇, intersecting AB at P. Justification: In ΔABA₇, PA₄ || BA₇. By Basic Proportionality Theorem, AP/PB = AA₄/A₄A₇. Since AA₄ = 4 units and A₄A₇ = 3 units (by construction), AP/PB = 4/3. Thus, P divides AB in the ratio 4:3. Final answer: The line segment is divided in the ratio 4:3, justified by BPT.
• Q: A line segment PQ is 6 cm long. Construct a point R on PQ such that PR = (2/5)PQ. Explain the ratio in which R divides PQ. A: Step 1: Draw a line segment PQ = 6 cm. Step 2: Draw a ray PX making an acute angle with PQ. Step 3: Mark 5 points P₁, P₂, P₃, P₄, P₅ on PX such that PP₁ = P₁P₂ = ... = P₄P₅. Step 4: Join Q to P₅. Step 5: Through P₂, draw a line parallel to QP₅, intersecting PQ at R. (This ensures PR:RQ = 2:3). Step 6: Explanation: If PR = (2/5)PQ, then PQ = PR + RQ = (5/2)PR. Substituting, (5/2)PR = PR + RQ. So, RQ = (5/2)PR - PR = (3/2)PR. Therefore, PR/RQ = PR / ((3/2)PR) = 2/3. Point R divides PQ in the ratio 2:3. Final answer: Point R divides PQ in the ratio 2:3.

Frequently Asked Questions