GANITA MANJARI · GRADE 9 · PART II · CHAPTER 9

PROVE — Converse of the Baudhāyana–Pythagoras Theorem

Think first. Use AI purposefully. Verify always.

01 · DISCOVER

Start by thinking.

A triangle has sides 8, 15 and 17. Does $8^{2}$ + $15^{2}$ = $17^{2}$ tell you something about an angle?

02 · UNDERSTAND

Converse of the Baudhāyana–Pythagoras Theorem

For positive side lengths a, b and c of a triangle, $a^{2}$ + $b^{2}$ = $c^{2}$ implies the angle opposite c is a right angle. To prove this, compare the given triangle with a constructed right triangle.

03 · KEY IDEAS

Keep the conditions clear.

In the forward theorem c is the hypotenuse.
Because a and b are positive, $c^{2}$ = $a^{2}$ + $b^{2}$ is greater than both $a^{2}$ and $b^{2}$ . Positive lengths therefore make c the longest side.
A constructed right triangle and SSS congruence establish the original angle; do not assume it in the diagram.

04 · VISUALISE

Follow the direction.

Original AIMS schematic, not to scale. Matching sides by SSS pairs A with X, B with Y and C with Z.

Given: ABC satisfies a² + b² = c². Its angle C is not assumed right.

05 · METHOD / REASONING

Make the argument checkable.

Construct perpendicular legs a and b. Use the forward theorem to find their hypotenuse. Match all three sides, then match the included angle by SSS.

06 · ORIGINAL AIMS WORKED EXAMPLE

Follow each step.

Given: BC = a, CA = b, AB = c and $a^{2}$ + $b^{2}$ = $c^{2}$ . Prove ∠C = 90°.

Construct ΔXYZ with YZ = a, ZX = b and ∠Z = 90°.
The forward theorem gives ${XY}^{2}$ = $a^{2}$ + $b^{2}$ = $c^{2}$ . Since lengths are positive, XY = c.
AB = XY, BC = YZ and CA = ZX, hence ΔABC ≅ ΔXYZ by SSS (A ↔ X, B ↔ Y, C ↔ Z).
Therefore ∠C = ∠Z = 90°. The triangle is right-angled opposite c.
Verify numerically: $8^{2}$ + $15^{2}$ = 64 + 225 = 289 = $17^{2}$ .

Formal AIMS proof

Given: Triangle ABC with BC = a, CA = b, AB = c; a, b, c are positive and $a^{2}$ + $b^{2}$ = $c^{2}$ .
To Prove: ∠C = 90°.
Construction: Construct triangle XYZ with YZ = a, ZX = b and ∠Z = 90°.
Property / Theorem: Baudhāyana–Pythagoras theorem; SSS congruence; equal corresponding angles.
Proof: ${XY}^{2}$ = $a^{2}$ + $b^{2}$ = $c^{2}$ . Both lengths are positive, so XY = c. Thus BC = YZ, CA = ZX and AB = XY. With A ↔ X, B ↔ Y, C ↔ Z, △ABC ≅ △XYZ by SSS. Hence ∠C = ∠Z = 90°.
Result: Triangle ABC is right-angled at C. No right angle in ABC was assumed.

07 · AIMS METHOD

Your turn to reason.

Attempt independently
Investigate the idea
Make connections
Scrutinise your reasoning

Do sides 9, 12, 15 form a right triangle? Justify the angle location.

Compare after your own attempt

Yes: $9^{2}$ + $12^{2}$ = 225 = $15^{2}$ . The right angle is opposite side 15.

08 · PRACTICE SPRINT

Use the idea in a new situation.

Ten original AIMS questions: 3 Warm-up, 4 Build, 2 Think and 1 Challenge.

Start Practice Sprint →

Check & Feedback: incorrect answers receive a conceptual hint and another attempt; correct answers include reasoning.

TEXTBOOK CONNECTION

Return to the source. Bring your reasoning.

Ganita Manjari · Grade 9 · Part II · Chapter 9. Use your textbook for the complete questions. These are topic references and original AIMS guidance, not reproduced exercises.

Textbook Guided Example

Example 5

Source page 4 · PROVE

The converse theorem and SSS

Construct a right triangle on the two shorter side lengths, then match all three sides.

AIMS reasoning checkpoint

Only the constructed triangle starts with a known right angle.

All textbook references →