Area of a triangle · 3 marks

Do you see why the two triangles fit together to make a parallelogram? (If you study the angles in the figure (e.g., ∠B’C’A’ and ∠BCA), you will see why this is so. Keep in mind the criterion by which we check whether two lines are parallel.)

Answer: Place the copy so that C′ is at A and A′ at C. Then ∠DAC = ∠B′C′A′ = ∠BCA, and these are alternate angles on AC, so AD ∥ BC. In the same way ∠DCA = ∠B′A′C′ = ∠BAC gives AB ∥ DC. Both pairs of opposite sides are parallel, so ABCD is a parallelogram.

Step-by-step solution

Idea: Two lines are parallel if a transversal makes equal alternate angles with them. The congruent copy supplies exactly those equal angles along the shared side AC.

Turn the copy A′B′C′ half round and put it on AC so that C′ falls on A and A′ falls on C (possible because C′A′ = CA). Call the position of B′ D.1 mark
Angles: ∠DAC = ∠B′C′A′ = ∠BCA (congruent triangles). These are alternate angles made by the transversal AC with lines AD and BC, so AD ∥ BC.1 mark
Similarly ∠DCA = ∠B′A′C′ = ∠BAC, alternate angles for lines DC and AB, so AB ∥ DC. A 4-gon with both pairs of opposite sides parallel is a parallelogram, so ABCD is a parallelogram (and the two triangles fit exactly).1 mark

With C′ on A and A′ on C, the equal angles ∠B′C′A′ = ∠BCA and ∠B′A′C′ = ∠BAC are alternate angles on AC, so AD ∥ BC and AB ∥ DC. Hence ABCD is a parallelogram.

Answer to write in the exam

Place △A′B′C′ with C′ on A, A′ on C; B′ at D.

∠DAC = ∠B′C′A′ = ∠BCA (△ABC ≅ △A′B′C′)

Alternate angles equal ⇒ AD ∥ BC

∠DCA = ∠B′A′C′ = ∠BAC ⇒ AB ∥ DC

∴ ABCD is a parallelogram.

Common mistakes that cost marks

Placing the copy without turning it, which does not give equal alternate angles.
Saying ‘opposite sides look parallel’ without naming the equal alternate angles.
Matching the wrong angles, e.g. ∠DAC with ∠BAC.

How this can come in the exam

Assertion–Reason (1 mark)

Assertion (A): The area of a triangle is half the area of a parallelogram with the same base and height.
Reason (R): Two congruent copies of a triangle can be fitted together to form a parallelogram with the same base and height.

Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A.
A is true but R is false.
A is false but R is true.

Show answer

(A) Both A and R are true, and R is the correct explanation of A.
Both true, and the parallelogram made from two equal triangles is the reason the triangle has half its area.

Try one yourself

A triangle has base 9 cm and height 4 cm. What is the area of the parallelogram made from two copies of it, and hence the area of the triangle?