Question

Below are given the measures of certain sides and angles of triangles. Identify those which cannot be constructed and, say why you cannot construct them. Construct rest of the triangle.

	Triangle	Given measurements
1.	ΔABC	m∠ A = 85°	m∠ B = 115°	AB = 5 cm
2.	ΔPQR	m∠ Q = 30°	m∠ R = 60°	QR = 4.7 cm
3.	ΔABC	m∠ A = 70°	m∠ B = 50°	AC = 3 cm
4.	ΔLMN	m∠ L = 60°	m∠ N = 120°	LM = 5 cm
5.	ΔABC	BC = 2 cm	AB = 4 cm	AC = 2 cm
6.	ΔPQR	PQ = 3.5 cm	QR = 4 cm	PR = 3.5 cm
7.	ΔXYZ	XY = 3 cm	YZ = 4 cm	XZ = 5 cm
8.	ΔDEF	DE = 4.5 cm	EF = 5.5 cm	DF = 4 cm

Answer 1

1. In Δ ABC, m∠ A = 85 °, m∠ B = 115 °, AB = 5 cm

Construction of Δ ABC is not possible because m∠ A = 85° + m∠ B = 200 °, and we know that the sum of angles of a triangle should be 180°.

2. To construct: Δ PQR where m∠ Q = 30° , m∠ R = 60° and QR = 4.7 cm.

Steps of construction:

(a) Draw a line segment QR = 4.7 cm.

(b) At point Q, draw ∠ XQR = 30 with the help of compass.

(c) At point R, draw ∠ YRQ = 60 with the help of compass.

(d) QX and RY intersect at point P.

It is the required triangle PQR.

3. We know that the sum of angles of a triangle is 180°.

Therefore  m∠ A + m∠ B + m∠ C = 180°

⇒ 70° + 50° + m∠ C = 180°

⇒ 120° + m∠ C = 180°

⇒ m∠ C = 180° – 120°

⇒ m∠ C = 60° 

(a) Draw a line segment AC = 3 cm.

(b) At point C, draw∠ YCA = 60° .

(c) At point A, draw ∠ XAC = 70°.

(d) Rays XA and YC intersect at point B It is the required triangle ABC.

4. In Δ LMN , m∠ L = 60 , m∠ N = 120 °, LM = 5 cm

This Δ LMN is not possible to construct because m∠ L + m∠ N = 60° + 120° = 180° which forms a linear pair. 

5. Δ ABC, BC = 2 cm, AB = 4 cm and AC = 2 cm

third side.  AB < BC + AC ⇒ 4 < 2 + 2

⇒ 4 = 4,

6. To construct: Δ PQR where PQ = 3.5 cm, QR = 4 cm and PR = 3.5 cm

Steps of construction:

(a) Draw a line segment QR = 4 cm.

(b) Taking Q as centre and radius 3.5 cm, draw an arc.

(c) Similarly, taking R as centre and radius 3.5 cm, draw an another arc which intersects the first arc at point P.

It is the required triangle PQR.

7. To construct: A triangle whose sides are XY = 3 cm, YZ = 4 cm and XZ = 5 cm.

Steps of construction:

(a) Draw a line segment ZY = 4 cm.

(b) Taking Z as centre and radius 5 cm, draw an arc.

(c) Taking Y as centre and radius 3 cm, draw another arc.

(d) Both arcs intersect at point X.

It is the required triangle XYZ.

8. To construct: A triangle DEF whose sides are DE = 4.5 cm, EF = 5.5 cm and DF = 4 cm.

Steps of construction:

(a) Draw a line segment EF = 5.5 cm.

(b) Taking E as centre and radius 4.5 cm, draw an arc.

(c) Taking F as centre and radius 4 cm, draw an another arc which intersects the first arc at point D.

It is the required triangle DEF.