Class 9 Ganita Manjari
Chapter 9: Propositions and their Converses
Part 5
[cite: 4]
Reversing Pythagorean Logic
We are very familiar with the BaudhΔyana-Pythagoras theorem: "Let \(a, b, c\) be the sidelengths of a triangle. If the triangle is right-angled, then \(a^2 + b^2 = c^2\)." [cite: 4]
But what if we start with the side lengths instead? If we find three lengths that satisfy \(a^2 + b^2 = c^2\), does that automatically guarantee the shape is a right triangle? [cite: 4] This is the Converse of the theorem. Let's explore how to mathematically prove that this converse is undeniably true by constructing a "twin" triangle. [cite: 4]
We are very familiar with the BaudhΔyana-Pythagoras theorem: "Let \(a, b, c\) be the sidelengths of a triangle. If the triangle is right-angled, then \(a^2 + b^2 = c^2\)." [cite: 4]
But what if we start with the side lengths instead? If we find three lengths that satisfy \(a^2 + b^2 = c^2\), does that automatically guarantee the shape is a right triangle? [cite: 4] This is the Converse of the theorem. Let's explore how to mathematically prove that this converse is undeniably true by constructing a "twin" triangle. [cite: 4]
1 Framing the Converse
Write down the exact wording for the Converse of the BaudhΔyana-Pythagoras theorem as formulated in the textbook.
Concept:
1
To write the converse, we keep the introductory sentence defining the variables exactly the same. We only swap the "If" condition and the "Then" conclusion. [cite: 4]
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Hint:
1
Original: "If the triangle is right-angled, then \(a^2 + b^2 = c^2\)." [cite: 4]
2
Swap the pieces around the "If" and "Then".
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Mistakes to Avoid:
!
Dropping the setup sentence: Don't just write "If $a^2+b^2=c^2$ then it's a right triangle." You must define what $a, b, c$ are first! "Let a, b, c be the sidelengths..." must remain at the front. [cite: 4]
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Solution:
The converse of the BaudhΔyana-Pythagoras theorem is formulated as follows:
"Let \(a, b, c\) be the sidelengths of a triangle. If \(a^2 + b^2 = c^2\), then the triangle is right-angled." [cite: 4]
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2 The Auxiliary Proof
Explain how the construction of an auxiliary triangle (\(\Delta XYZ\)) is used to mathematically prove that the converse of the BaudhΔyana-Pythagoras theorem is true.
Concept:
1
Since we don't know if $\Delta ABC$ is right-angled, we purposely build a *new* triangle ($\Delta XYZ$) that is guaranteed to be right-angled, using the leg lengths $a$ and $b$. [cite: 4]
2
We then use algebra to prove the hypotenuse of the new triangle must equal $c$. Finally, we use SSS Congruence to prove the two triangles are identical, meaning $\Delta ABC$ inherited the right angle! [cite: 4]
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Hint:
1
Because \(\Delta XYZ\) is right-angled, we can apply the original Pythagoras theorem: \(XY^2 = a^2 + b^2\). [cite: 4]
2
Since we are given \(a^2 + b^2 = c^2\) for the original triangle, what does \(XY\) equal? [cite: 4]
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Mistakes to Avoid:
!
Assuming $\Delta ABC$ is a right triangle at the start: If you start by saying $\Delta ABC$ is right-angled, you are engaging in circular logic. You must only assume $a^2 + b^2 = c^2$ and *prove* the right angle via $\Delta XYZ$. [cite: 4]
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Solution:
1. We start with \(\Delta ABC\) having sides \(a, b, c\) where \(a^2 + b^2 = c^2\). [cite: 4]
2. We intentionally construct a new right-angled triangle, \(\Delta XYZ\), where the perpendicular legs \(YZ\) and \(XZ\) have lengths \(a\) and \(b\). [cite: 4]
3. Because \(\Delta XYZ\) is a right triangle, the BaudhΔyana-Pythagoras theorem applies directly to it: \(XY^2 = a^2 + b^2\). [cite: 4]
4. Substituting the given condition from the first triangle (\(a^2 + b^2 = c^2\)), we get \(XY^2 = c^2\), which means the hypotenuse \(XY = c\). [cite: 4]
5. Now, both \(\Delta ABC\) and \(\Delta XYZ\) have exact matching side lengths of \(a, b,\) and \(c\). By the SSS (Side-Side-Side) criterion, the two triangles are congruent. [cite: 4]
6. Because they are congruent, their angles match. Since \(\Delta XYZ\) has a right angle, \(\Delta ABC\) must also have a right angle, proving the converse is True! [cite: 4]
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Objective 1: Congruence Criterion
Which congruence criterion is specifically used in the proof to conclude that \(\Delta ABC\) is congruent to \(\Delta XYZ\)?
Concept:
1
When comparing the two triangles, we know side $a$ matches side $a$, side $b$ matches side $b$, and we calculated that side $c$ matches side $c$. [cite: 4]
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Hint:
1
Which congruence rule relies exclusively on the lengths of the three boundaries of a triangle? [cite: 4]
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Mistakes to Avoid:
!
Picking RHS prematurely: While $\Delta XYZ$ is a right triangle, we don't *know* $\Delta ABC$ is a right triangle yet (that's what we are trying to prove!). Therefore, we cannot use Right-Angle Hypotenuse Side (RHS) to prove they are identical. We must rely on SSS. [cite: 4]
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Solution:
In the proof, we established that \(\Delta ABC\) has sides \(a, b, c\). [cite: 4]
We constructed \(\Delta XYZ\) with legs \(a, b\) and calculated its hypotenuse to also be \(c\). [cite: 4]
Because all three corresponding sides are identical in length, we use the SSS criterion for congruence. [cite: 4]
Correct Option: (B)
Objective 2: Verifying the Converse Truth
Based on the mathematical proof using \(\Delta XYZ\), what is the final conclusion regarding the converse of the BaudhΔyana-Pythagoras Theorem?
Concept:
1
Because $\Delta ABC$ and $\Delta XYZ$ are congruent, every single property matches perfectly. Since $\Delta XYZ$ was explicitly constructed to have a $90^\circ$ angle, $\Delta ABC$ must have one too! [cite: 4]
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Hint:
1
Read the final sentence of Example 5 in the text: "Hence the converse of the BaudhΔyana-Pythagoras Theorem is..." [cite: 4]
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Mistakes to Avoid:
!
Assuming converses are always false: While the rain/road example showed a false converse, this geometric proof shows that mathematical converses are very often True! [cite: 4]
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Solution:
The text concludes the proof by stating: "We conclude that \(\Delta ABC\) had to be a right triangle after all. Hence the converse of the BaudhΔyana-Pythagoras Theorem is also true!" [cite: 4]
Correct Option: (B)
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Extra Practice 1
You are given a triangle with side lengths of 5 cm, 12 cm, and 13 cm. Use the converse of the BaudhΔyana-Pythagoras theorem to mathematically prove whether this is a right-angled triangle or not. Show your calculations.
Concept:
1
Because we know the converse is true, we can simply plug the three side lengths into the equation $a^2 + b^2 = c^2$. [cite: 4] If the two sides of the equation balance perfectly, it is guaranteed to be a right triangle!
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Hint:
1
The longest side must always be the hypotenuse (\(c\)). Set \(c = 13\).
2
Calculate \(5^2 + 12^2\). Does it exactly equal \(13^2\)?
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Mistakes to Avoid:
!
Mixing up the hypotenuse: If you calculate $12^2 + 13^2 = 5^2$, the equation will fail. Always ensure the largest number is isolated on one side of the equals sign!
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Solution:
1. Identify the variables: The longest side is 13, so it must act as the hypotenuse (\(c\)). Let \(a = 5\) and \(b = 12\).
2. Apply the equation: Check if \(a^2 + b^2 = c^2\). [cite: 4]
\(5^2 + 12^2 = 25 + 144 = 169\).
\(13^2 = 169\).
3. Conclusion: Because \(169 = 169\), the condition is perfectly satisfied. According to the converse of the BaudhΔyana-Pythagoras theorem, this triangle is right-angled! [cite: 4]
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