Understanding the Problem

The angles of a triangle are in the ratio 1:2:3, and the hypotenuse is 16 cm. We need to find the sum of the other two sides.

Step 1: Determine the Angles

Let the angles be $x$, $2x$, and $3x$ degrees.

Sum of angles in a triangle is 180°:

$x+2x+3x=180$

$6x=180$

$x=30$

So the angles are 30°, 60°, and 90°.

Step 2: Identify the Triangle Type

A triangle with angles 30°, 60°, and 90° is a right triangle. The side opposite the 90° angle is the hypotenuse.

Step 3: Recall the Side Ratios

In a 30-60-90 triangle, the sides are in the ratio:

Side opposite 30° : Side opposite 60° : Hypotenuse = $1:\sqrt{3}:2$

Let the sides be:

Opposite 30° = $a$

Opposite 60° = $a\sqrt{3}$

Hypotenuse = $2a$

Step 4: Find the Value of $a$

Given hypotenuse = 16 cm:

$2a=16$

$a=8$

Step 5: Find the Other Two Sides

Side opposite 30° = $a=8 \mathrm{cm}$

Side opposite 60° = $a\sqrt{3}=8\sqrt{3} \mathrm{cm}$

Step 6: Find the Sum of the Other Two Sides

Sum = $8+8\sqrt{3}$

Final Answer

The sum of the other two sides is $8+8\sqrt{3}$ cm.

Related Topics and Formulae

Properties of Triangles

• Sum of interior angles = 180°
• Pythagorean Theorem: In a right triangle, $a^2+b^2=c^2$, where $c$ is the hypotenuse.

Special Right Triangles

• 30-60-90 Triangle: Sides are in ratio $1:\sqrt{3}:2$
• 45-45-90 Triangle: Sides are in ratio $1:1:\sqrt{2}$