We are given an isosceles triangle with a base of 10 cm and an altitude of 12 cm. An isosceles triangle is defined as a triangle that has two sides of equal length. The altitude to the base of an isosceles triangle is a perpendicular line segment from the vertex opposite the base to the base. This altitude has important properties that help us solve this problem.

Understanding the Isosceles Triangle and its Altitude

In an isosceles triangle, the altitude drawn to the base does more than just provide the height; it also bisects the base. This means it divides the base into two segments of equal length. Furthermore, this altitude splits the isosceles triangle into two congruent right-angled triangles.

Let's consider our specific triangle:

• The base is 10 cm.
• The altitude is 12 cm.

When the altitude is drawn to the base, it creates two right-angled triangles. Each of these right-angled triangles has:

• One leg equal to half of the base of the isosceles triangle.
• The other leg equal to the altitude of the isosceles triangle.
• The hypotenuse equal to one of the equal sides of the isosceles triangle.

The length of half the base is $\frac{10 \text{ cm}}{2} = 5 \text{ cm}$.

So, for each right-angled triangle, we have:

• Leg 1 = 5 cm (half base)
• Leg 2 = 12 cm (altitude)
• Hypotenuse = length of the equal side (unknown)

Applying the Pythagorean Theorem

To find the length of the hypotenuse in a right-angled triangle, we use the Pythagorean theorem. This theorem states that the square of the length of the hypotenuse ($c$) is equal to the sum of the squares of the lengths of the other two sides ($a$ and $b$). The formula is: $a^2 + b^2 = c^2$.

In our case, the legs are 5 cm and 12 cm, and the hypotenuse is the equal side we want to find. Let $s$ represent the length of the equal side.

Using the Pythagorean theorem:

$\text{Leg 1}^2 + \text{Leg 2}^2 = \text{Hypotenuse}^2$

$(5 \text{ cm})^2 + (12 \text{ cm})^2 = s^2$

$25 \text{ cm}^2 + 144 \text{ cm}^2 = s^2$

$169 \text{ cm}^2 = s^2$

To find $s$, we take the square root of both sides:

$s = \sqrt{169 \text{ cm}^2}$

$s = 13 \text{ cm}$

So, the length of each equal side of the isosceles triangle is 13 cm.

Final Answer Determination

By using the property of the altitude in an isosceles triangle and applying the Pythagorean theorem to the resulting right-angled triangles, we calculated the length of the equal side. The calculation shows that the length is 13 cm.

Revision Table: Key Concepts for Isosceles Triangle Problems

Additional Information: Beyond Isosceles Triangles

Triangle geometry involves many shapes and formulas. Here are some related concepts that are often encountered:

• Scalene Triangle: All sides and angles are different.
• Equilateral Triangle: All sides are equal, all angles are 60°. Altitude, median, and angle bisector from any vertex are the same line segment.
• Median: A line segment from a vertex to the midpoint of the opposite side.
• Area of a Triangle: Calculated as $\frac{1}{2} \times \text{base} \times \text{height}$ for any triangle.
• Special Right Triangles: Like 3-4-5 or 5-12-13 triangles (Pythagorean triples) where side lengths are integers. Our problem involves a 5-12-13 triangle.

Understanding these concepts helps in tackling a wider range of geometry problems.