Question

Gopi constructed a right-angled triangle. By modifying the dimensions of first triangle, he drew another triangle. The modifications are, the largest side is increased by 5 cm, the smallest side is doubled and the third side is increased by 50%. If the triangle formed with these new dimensions has equal angles, then what is the perimeter of the new triangle?

• A. 45 cm
• B. 60 cm
• C. 90 cm
• D. 120 cm

Hint:

Equal angles means the new triangle is equilateral, so all three new sides must be equal.

Answer 1

The Correct Option is C

Step 1: Set up the original triangle.
Let the smallest side be \( a \), the third side be \( b \), and the largest side (hypotenuse) be \( c \).
Since it is right angled, \( a^2 + b^2 = c^2 \).

Step 2: Write the new sides.
New smallest side \( = 2a \), new third side \( = 1.5b \), new largest side \( = c + 5 \).

Step 3: Interpret equal angles.
A triangle whose three angles are all equal to each other must be equilateral, so its three sides are equal.
So \( 2a = 1.5b = c + 5 = s \), where \( s \) is the common new side.

Step 4: Express a, b, c in terms of s.
\( a = \dfrac{s}{2} \), \( b = \dfrac{2s}{3} \), \( c = s - 5 \).

Step 5: Substitute into the right angle relation.
\( \left(\dfrac{s}{2}\right)^2 + \left(\dfrac{2s}{3}\right)^2 = (s-5)^2 \)
\( \dfrac{9s^2}{36} + \dfrac{16s^2}{36} = s^2 - 10s + 25 \)
\( \dfrac{25s^2}{36} = s^2 - 10s + 25 \), which simplifies to \( 11s^2 - 360s + 900 = 0 \).

Step 6: Solve the quadratic.
The discriminant is \( 360^2 - 4 \times 11 \times 900 = 90000 \), and \( \sqrt{90000} = 300 \).
\( s = \dfrac{360 + 300}{22} = 30 \) or \( s = \dfrac{360 - 300}{22} = \dfrac{30}{11} \), which is too small to keep \( c = s - 5 \) positive and valid.
So \( s = 30 \).

Step 7: Check with the original sides.
\( a = 15 \), \( b = 20 \), \( c = 25 \), and \( 15^2 + 20^2 = 225 + 400 = 625 = 25^2 \), a valid right triangle.
New sides: \( 2(15) = 30 \), \( 1.5(20) = 30 \), \( 25 + 5 = 30 \), all equal, confirming the equilateral triangle.

Final Answer:
Perimeter \( = 3 \times 30 = 90 \) cm. \[ \boxed{90 \text{ cm}} \]