Understanding the Problem

The angles of a triangle are in the ratio 1 : 1 : 2. Let the angles be $x$, $x$, and $2x$ degrees. Since the sum of angles in a triangle is 180°, we can write:

$x+x+2x=180$

$4x=180$

$x=45$

Therefore, the angles are 45°, 45°, and 90°. This is an isosceles right triangle.

Step 1: Identify the Triangle Type and Sides

In a 45°-45°-90° triangle, the sides are in the ratio $1:1:\sqrt{2}$. This means the two legs are equal, and the hypotenuse is $\sqrt{2}$ times the length of a leg.

Step 2: Determine Which Side is 15 cm

The problem states "one of the sides is 15cm" but does not specify if it is a leg or the hypotenuse. We must consider both cases to find which one leads to a valid perimeter that matches one of the options.

Case 1: The given side (15 cm) is a leg.

If one leg = 15 cm, then:

Other leg = 15 cm

Hypotenuse = $15\sqrt{2}$ cm

Perimeter, $P=15+15+15\sqrt{2}=30+15\sqrt{2}=15(2+\sqrt{2})$ cm

This matches the fourth option: $15(2+\sqrt{2})$

Case 2: The given side (15 cm) is the hypotenuse.

If hypotenuse = 15 cm, then:

Leg = $\frac{15}{\sqrt{2}}=\frac{15}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}=\frac{15}{\sqrt{2}}$ cm (This is a valid length, but let's check the perimeter).

Perimeter, $P=\frac{15}{\sqrt{2}}+\frac{15}{\sqrt{2}}+15=15+\frac{30}{\sqrt{2}}=15+15\sqrt{2}$ cm

This simplifies to $15(1+\sqrt{2})$ cm, which is not one of the given options. Therefore, Case 1 is the correct interpretation.

Final Answer

The perimeter of the triangle is $15(2+\sqrt{2})$ cm.

Related Topics & Formulae

Angle Sum Property of a Triangle: The sum of the three interior angles of any triangle is always 180°.

Properties of Special Triangles:

• 45°-45°-90° Triangle (Isosceles Right Triangle): The sides are in the ratio $1:1:\sqrt{2}$.
• 30°-60°-90° Triangle: The sides are in the ratio $1:\sqrt{3}:2$.

Pythagorean Theorem: In any right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the legs (a and b).

$a^2+b^2=c^2$