Understanding the Triangle and its Altitudes

The problem asks for the length of the smallest altitude of a triangle with side lengths 5 cm, 12 cm, and 13 cm.

First, let's identify the type of triangle. We can check if these side lengths form a right-angled triangle using the Pythagorean theorem ($a^2 + b^2 = c^2$).

Let the side lengths be $a=5$, $b=12$, and $c=13$.

Calculate the sum of the squares of the two shorter sides:

\begin{equation*} 5^2 + 12^2 = 25 + 144 = 169 \end{equation*}

Calculate the square of the longest side:

\begin{equation*} 13^2 = 169 \end{equation*}

Since $5^2 + 12^2 = 13^2$, the triangle is a right-angled triangle. The sides 5 cm and 12 cm are the legs, and the side 13 cm is the hypotenuse.

Relating Altitudes to Side Lengths

In any triangle, the altitude corresponding to a specific side is the perpendicular distance from the opposite vertex to that side. The area of a triangle can be calculated using the formula:

\begin{equation*} \text{Area} = \frac{1}{2} \times \text{base} \times \text{altitude} \end{equation*}

For a fixed area, if the base is larger, the corresponding altitude must be smaller, and vice versa. Therefore, the smallest altitude corresponds to the longest side of the triangle.

In our right-angled triangle:

• The altitude corresponding to the side of length 5 cm is the side of length 12 cm (as they are perpendicular legs).
• The altitude corresponding to the side of length 12 cm is the side of length 5 cm (as they are perpendicular legs).
• The altitude corresponding to the side of length 13 cm (the hypotenuse) is the perpendicular segment from the right-angle vertex to the hypotenuse. This will be the smallest altitude because 13 cm is the longest side.

Calculating the Area of the Triangle

We can easily calculate the area of the right-angled triangle using the two legs as the base and height:

\begin{equation*} \text{Area} = \frac{1}{2} \times \text{leg}_1 \times \text{leg}_2 \end{equation*}

\begin{equation*} \text{Area} = \frac{1}{2} \times 5 \text{ cm} \times 12 \text{ cm} \end{equation*}

\begin{equation*} \text{Area} = \frac{1}{2} \times 60 \text{ cm}^2 \end{equation*}

\begin{equation*} \text{Area} = 30 \text{ cm}^2 \end{equation*}

Finding the Smallest Altitude Length

The smallest altitude ($h$) is the altitude corresponding to the longest side (the hypotenuse, 13 cm). We can use the area formula again with the hypotenuse as the base:

\begin{equation*} \text{Area} = \frac{1}{2} \times \text{hypotenuse} \times h \end{equation*}

Substitute the calculated area and the hypotenuse length:

\begin{equation*} 30 \text{ cm}^2 = \frac{1}{2} \times 13 \text{ cm} \times h \end{equation*}

Now, solve for $h$:

\begin{equation*} 2 \times 30 \text{ cm}^2 = 13 \text{ cm} \times h \end{equation*}

\begin{equation*} 60 \text{ cm}^2 = 13 \text{ cm} \times h \end{equation*}

\begin{equation*} h = \frac{60 \text{ cm}^2}{13 \text{ cm}} \end{equation*}

\begin{equation*} h = \frac{60}{13} \text{ cm} \end{equation*}

Let's calculate the value and round it to one decimal place:

\begin{equation*} \frac{60}{13} \approx 4.61538... \end{equation*}

Rounding to one decimal place, the smallest altitude is approximately 4.6 cm.

Comparing with Options

The calculated smallest altitude is approximately 4.6 cm. Comparing this with the given options:

The calculated value matches Option 4.

Revision Table: Key Concepts for Triangle Altitudes

Additional Information on Triangle Properties

• In an acute-angled triangle, all altitudes lie inside the triangle.
• In an obtuse-angled triangle, two altitudes lie outside the triangle.
• In a right-angled triangle, the two legs are the altitudes corresponding to the acute-angled vertices, and they meet at the right-angle vertex. The third altitude is from the right-angle vertex to the hypotenuse.
• The point where the three altitudes of a triangle intersect is called the orthocenter.
• For any triangle, if $s_1 < s_2 < s_3$ are the side lengths and $h_1, h_2, h_3$ are the corresponding altitudes, then $h_1 > h_2 > h_3$. The shortest altitude is opposite the longest side.