Proving the Triangle is Right Angled

Video Explanation

Question

In triangle ABC: \[ \angle A = x^\circ,\quad \angle B = 3x^\circ,\quad \angle C = y^\circ \] Given: \[ 3y – 5x = 30 \] Prove that the triangle is right angled.

Solution

Step 1: Use Angle Sum Property

\[ x + 3x + y = 180 \]

\[ 4x + y = 180 \quad (1) \]

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Step 2: Use Given Condition

\[ 3y – 5x = 30 \quad (2) \]

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Step 3: Solve Linear Equations

From (1):

\[ y = 180 – 4x \]

Substitute into (2):

\[ 3(180 – 4x) – 5x = 30 \]

\[ 540 – 12x – 5x = 30 \]

\[ 540 – 17x = 30 \]

\[ 17x = 510 \Rightarrow x = 30 \]

Then:

\[ y = 180 – 4(30) = 60 \]

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Step 4: Find Angles

\[ \angle A = 30^\circ \]

\[ \angle B = 3x = 90^\circ \]

\[ \angle C = 60^\circ \]

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Step 5: Conclusion

Since one angle of the triangle is \(90^\circ\),

\[ \boxed{\text{The triangle is right angled}} \]

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Verification

Sum: \(30 + 90 + 60 = 180^\circ\) ✔