Understanding Triangle Angles in Ratio

The question asks to identify the type of triangle when its angles are in the ratio 1 ∶ 2 ∶ 3. To solve this, we first need to find the actual measures of the three angles in the triangle.

Calculating the Angles of the Triangle

The sum of the interior angles in any triangle is always 180 degrees. If the angles are in the ratio 1 ∶ 2 ∶ 3, we can represent the angles as multiples of a common value, let's call it \(x\).

• The first angle is \(1x\) or simply \(x\).
• The second angle is \(2x\).
• The third angle is \(3x\).

Using the property that the sum of angles in a triangle is 180 degrees, we can set up the equation:

\(x + 2x + 3x = 180^\circ\)

Combine the terms on the left side:

\(6x = 180^\circ\)

Now, solve for \(x\) by dividing both sides by 6:

\(x = \frac{180^\circ}{6}\)

\(x = 30^\circ\)

Now that we have the value of \(x\), we can find the measure of each angle:

• First angle: \(x = 30^\circ\)
• Second angle: \(2x = 2 \times 30^\circ = 60^\circ\)
• Third angle: \(3x = 3 \times 30^\circ = 90^\circ\)

The three angles of the triangle are \(30^\circ\), \(60^\circ\), and \(90^\circ\).

Identifying the Type of Triangle

Now we need to determine the type of triangle based on these angles. Let's look at the options provided:

• Isosceles triangle: A triangle with at least two angles of equal measure. Our angles are \(30^\circ, 60^\circ, 90^\circ\), which are all different. So, it's not an isosceles triangle (unless it's also equilateral, which it isn't).
• Equilateral triangle: A triangle with all three angles of equal measure (\(60^\circ\) each). Our angles are \(30^\circ, 60^\circ, 90^\circ\), which are not all equal. So, it's not an equilateral triangle.
• Right-angle triangle: A triangle that has one angle measuring exactly 90 degrees. Our calculated angles are \(30^\circ, 60^\circ, 90^\circ\), and one of these angles is indeed \(90^\circ\). This fits the definition of a right-angle triangle.
• Obtuse-angle triangle: A triangle that has one angle measuring more than 90 degrees. Our angles are \(30^\circ, 60^\circ, 90^\circ\). None of these angles are greater than 90 degrees. So, it's not an obtuse-angle triangle.

Since the triangle has an angle of \(90^\circ\), it is a right-angle triangle.

Summary of Angle Calculation

The angles are \(30^\circ, 60^\circ, 90^\circ\). A triangle with a \(90^\circ\) angle is classified as a right-angle triangle.

Revision Table: Triangle Types by Angles

Additional Information: Properties of Right-Angle Triangles

A right-angle triangle has several special properties:

• It contains one angle that is exactly \(90^\circ\).
• The side opposite the right angle is called the hypotenuse, which is the longest side of the triangle.
• The other two sides are called legs or cathetus.
• The sum of the other two angles (the acute angles) is always \(90^\circ\). In our case, \(30^\circ + 60^\circ = 90^\circ\).
• The Pythagorean theorem applies: the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides (\(a^2 + b^2 = c^2\)).

The triangle with angles in the ratio 1:2:3 is a specific type of right-angle triangle, often referred to as a 30-60-90 triangle, which has unique side length ratios.