Construction of Triangles: Advanced Cases

The construction of triangles is a fundamental concept in geometry. Given specific measurements, such as the base, base angles, and the sum or difference of the other two sides, or the perimeter and base angles, we can accurately construct a triangle using a compass and straightedge (ruler). This involves understanding and applying various geometric principles to achieve a unique and accurate representation of the triangle.

The key is to translate the given information into a series of geometric steps, ensuring the correct lengths and angles are maintained.

Formulae

While there isn’t a single formula to construct a triangle, the process relies on geometric principles and properties. Here are some important related concepts:

• Sum of Interior Angles: The sum of the interior angles of a triangle is always $180^\circ$.
• Angle Sum Property: $ \angle A + \angle B + \angle C = 180^\circ $ (where A, B, and C are the angles of the triangle).
• Relationship between sides and angles: The side opposite a larger angle is always longer than the side opposite a smaller angle.

Examples

Example-1: Construction when base, base angle and sum of the other two sides is given.

1. Given: Base $BC = 6$ cm, $\angle B = 60^\circ$, and $AB + AC = 9$ cm.
2. Steps:
   1. Draw the base $BC = 6$ cm.
   2. At B, construct $\angle XBC = 60^\circ$.
   3. Cut a line segment $BD = 9$ cm from BX. ($BD = AB + AC$)
   4. Join DC.
   5. Draw the perpendicular bisector of $DC$, intersecting $BD$ at $A$.
   6. Join $AC$.
   7. Therefore, $\triangle ABC$ is the required triangle. (Since $AD = AC$)

Example-2: Construction when base, base angle and difference of the other two sides is given.

1. Given: Base $BC = 5$ cm, $\angle B = 45^\circ$, and $AB – AC = 2$ cm (or $AC – AB = 2$ cm).
2. Steps (for AB – AC = 2 cm):
   1. Draw the base $BC = 5$ cm.
   2. At B, construct $\angle XBC = 45^\circ$.
   3. Cut a line segment $BD = 2$ cm from BX.
   4. Join DC.
   5. Draw the perpendicular bisector of $DC$, intersecting $BX$ at $A$.
   6. Join $AC$.
   7. Therefore, $\triangle ABC$ is the required triangle.
3. Steps (for AC – AB = 2 cm):
   1. Draw the base $BC = 5$ cm.
   2. At B, construct $\angle XBC = 45^\circ$.
   3. Extend $BC$ to $X$.
   4. Cut a line segment $BD = 2$ cm from BX.
   5. Join DC.
   6. Draw the perpendicular bisector of $DC$, intersecting $BA$ at $A$.
   7. Join $AC$.
   8. Therefore, $\triangle ABC$ is the required triangle.

Theorem with Proof

Theorem: The sum of any two sides of a triangle is greater than the third side.

Proof:

Consider a triangle $ABC$. We need to prove that:

• $AB + AC > BC$
• $AB + BC > AC$
• $AC + BC > AB$

To prove the first inequality, extend $BA$ to point $D$ such that $AD = AC$. Join $CD$.

In $\triangle ACD$, $AD = AC$. Therefore, $\angle ACD = \angle ADC$ (angles opposite equal sides are equal).

Now, $\angle BCD > \angle ADC$ (since $\angle BCD$ is the sum of $\angle BCA$ and $\angle ACD$).

Therefore, in $\triangle BCD$, $\angle BCD > \angle BDC$. Hence, $BD > BC$ (side opposite larger angle is larger).

But $BD = BA + AD = BA + AC$.

So, $AB + AC > BC$.

Similarly, we can prove the other inequalities by extending different sides and constructing the angles.

Common mistakes by students

• Incorrect Angle Measurement: Using the protractor incorrectly leads to inaccurate angles.
• Incorrect Line Segment Measurement: Not using the ruler or compass accurately when cutting line segments.
• Perpendicular Bisector Construction: Incorrectly drawing the perpendicular bisector, leading to incorrect point location.
• Confusing Sum and Difference of Sides: Not correctly applying the construction steps depending on whether the sum or difference of the sides is given.

Real Life Application

Triangle construction is fundamental in many areas, including:

• Architecture and Engineering: Designing buildings, bridges, and other structures.
• Surveying: Determining distances and areas of land.
• Navigation: Calculating positions using triangulation.
• Carpentry and Construction: Creating precise angles for framing and other woodwork.

Fun Fact

The ancient Egyptians used rope with knots tied at equal intervals to create right angles, a fundamental step in their construction of pyramids. This is a practical application of the 3-4-5 right triangle, a special case which could be constructed by dividing rope into segments with the mentioned proportion.

Recommended YouTube Videos for Deeper Understanding

Q.1 Construct a triangle $ABC$ given $BC = 6$ cm, $\angle B = 60^\circ$, and $AB + AC = 10$ cm. Which of the following steps is the correct first step?

Check Solution

Ans: A

The first step involves drawing the base and one angle.

Q.2 Construct a triangle $PQR$ with perimeter 12 cm and $\angle Q = 50^\circ$ and $\angle R = 70^\circ$. What is the length of the base $QR$?

Check Solution

Ans: B

We are given the perimeter and the angles. Base length is not a direct outcome.

Q.3 Construct a triangle $XYZ$ given $YZ = 5$ cm, $\angle Y = 75^\circ$, and $XY – XZ = 2$ cm. What is the second step after drawing $YZ$?

Check Solution

Ans: A

The second step is to draw the ray representing the angle.

Q.4 In constructing a triangle $ABC$ given $BC = 7$ cm, $\angle B = 45^\circ$, and $AB – AC = 1$ cm, where will point $D$ lie if $BD = AB$?

Check Solution

Ans: A

Since $AB – AC$ is given and $AB>AC$, $D$ will be on the extension.

Q.5 For constructing a triangle with a perimeter of 15 cm and base angles $60^\circ$ and $45^\circ$, after drawing the base segment and drawing the angles, what is the next step?

Check Solution

Ans: C

Constructing the triangle based on angles requires finding the intersection.

Next Topic: Area of a Triangle using Heron’s Formula