Proposition P: If two triangles have the same area, then they are congruent.
Converse Q: If two triangles are congruent, then they have the same area.
Determine if these statements are true or not. Justify the true statements and give a counterexample for each false statement.
Step-by-step solution
Idea: Area = ½ × base × height, and many different base–height pairs give the same product, so equal area does not fix the shape. Congruent triangles, on the other hand, are exact copies of each other.
- P claims that equal area forces congruence. Look for two triangles with equal areas but different shapes.½ mark
- Triangle 1: right triangle with legs 6 cm and 4 cm. Area = ½ × 6 × 4 = 12 cm2.
Triangle 2: right triangle with legs 8 cm and 3 cm. Area = ½ × 8 × 3 = 12 cm2.½ mark - Triangle 1 has sides 4 cm, 6 cm and √52 ≈ 7.2 cm (its longest side). Triangle 2 has a side of 8 cm, longer than every side of Triangle 1. Congruent triangles have all three sides equal, so these two are not congruent.½ mark
- So P is false; this pair of triangles is a counterexample.½ mark
- Q: if two triangles are congruent, one can be placed exactly on the other, with all sides and angles matching. They then cover exactly the same region, so their areas are equal. Q is true.½ mark
- So here the converse Q is true while the proposition P is false.½ mark
Check: Hypotenuses: √(36 + 16) = √52 ≈ 7.21 cm and √(64 + 9) = √73 ≈ 8.54 cm, so the three sides really are different (4, 6, 7.21 and 3, 8, 8.54).
Answer to write in the exam
P: Take right triangles with legs 6 cm, 4 cm and 8 cm, 3 cm.
Areas: ½ × 6 × 4 = 12 cm2; ½ × 8 × 3 = 12 cm2
Sides of the first: 4 cm, 6 cm, √52 ≈ 7.2 cm; the second has a side 8 cm.
Equal areas but not congruent. ∴ P is false.
Q: Congruent triangles coincide when placed on each other, so they cover the same region.
∴ Q is true: congruent triangles have the same area.
Common mistakes that cost marks
- Giving two triangles that are actually congruent (for example the same triangle turned round) as the counterexample.
- Justifying Q with one numerical case. Q must hold for all congruent triangles: they coincide, so they cover the same region.
- Thinking equal areas mean equal bases and equal heights. Many base–height pairs give the same area.
How this can come in the exam
Which of these statements is true?
- Triangles with equal perimeters are congruent.
- Triangles with equal areas are congruent.
- Congruent triangles have equal perimeters.
- Two triangles that each have two equal sides are congruent.
Show answer
(C) Congruent triangles have equal perimeters.
(C) is true: congruent triangles have all three sides equal, so equal perimeters. (A) fails: sides 3, 4, 5 and 4, 4, 4 both have perimeter 12. (B) fails: right triangles with legs 6, 4 and 8, 3 both have area 12. (D) fails: sides 5, 5, 2 and 5, 5, 8.
Give two rectangles that have the same area but are not congruent. Is ‘congruent rectangles have the same perimeter’ true?
Show answer
9 cm × 4 cm and 6 cm × 6 cm both have area 36 cm2, but their sides differ, so they are not congruent (1 mark). True: congruent rectangles have equal sides, so equal perimeters (1 mark).Try one yourself
Show that ‘If two triangles have the same perimeter, then they are congruent’ is false.
Show answer
Sides 5 cm, 5 cm, 6 cm and 4 cm, 6 cm, 6 cm: both perimeters are 16 cm, but the first has two 5 cm sides and the second has none, so they are not congruent.
More questions like this
- It can also happen that both the proposition and converse are false! Can you give an example?
- Here is a statement of the Baudhāyana–Pythagoras theorem.
Let a, b, c be the sidelengths of a triangle. If the triangle is right-angled, then a2 + b2 = c2.
What is its converse? Is this converse true? - 1. Identify real-life examples in which a proposition is true but not the converse. Give nice counterexamples!
2. Give more examples from geometry as well as number theory in which a proposition is true but not its converse. Give appropriate counterexamples. - If two lines are parallel, then the corresponding angles formed by a transversal are equal.
- If a quadrilateral is a square, then all its angles are equal.