An isosceles triangle has side lengths of \(13\) cm, \(13\) cm, and \(10\) cm. Find the length of the altitude to the base of \(10\) cm.
["# Finding the Altitude of an Isosceles Triangle with Side Lengths 13 cm, 13 cm, and 10 cm", "An isosceles triangle is defined as a triangle with two equal side lengths and a distinct base. In this case, the triangle has two sides measuring 13 cm each, and the base measuring 10 cm. To fully understand its geometric properties, especially the length of the altitude drawn to the base, let’s analyze the triangle step-by-step.", "## Understanding the Isosceles Triangle Configuration", "Given:\n- Legs: ( AC = AB = 13 ) cm\n- Base: ( BC = 10 ) cm", "Because the triangle is isosceles with equal legs ( AB = AC ), the altitude drawn from vertex ( A ) to base ( BC ) will bisect the base into two equal segments. This creates two right-angled triangles within the original triangle, each with hypotenuse ( 13 ) cm, base ( \frac{10}{2} = 5 ) cm, and unknown height ( h ), which is the altitude we want to find.", "## Applying the Pythagorean Theorem", "Each right triangle formed has:\n- Hypotenuse = 13 cm\n- One leg = 5 cm (half of the base)\n- Other leg = altitude ( h )", "Using the Pythagorean theorem:\n[\nh^2 + 5^2 = 13^2\n]\n[\nh^2 + 25 = 169\n]\n[\nh^2 = 169 - 25 = 144\n]\n[\nh = \sqrt{144} = 12 \ ext{ cm}\n]", "## Conclusion", "The length of the altitude from the apex to the base of the isosceles triangle with sides 13 cm, 13 cm, and 10 cm is 12 cm. This altitude not only confirms the triangle’s symmetry but also plays a key role in calculations involving area, perimeter, and overall shape properties.", "For anyone working with isosceles triangles—whether for geometry class, architectural design, or engineering applications—knowing how to calculate the altitude efficiently is essential. Use this method whenever you need to find heights from the apex in symmetric triangles.", "---", "Keywords: isosceles triangle altitude, find altitude of 13-13-10 triangle, area calculation isosceles triangle, geometry problem solution, height of isosceles triangle, Pythagorean theorem isosceles, triangle altitude formula", "Meta Description: Learn how to find the altitude of an isosceles triangle with sides 13 cm, 13 cm, and 10 cm using the Pythagorean theorem. Step-by-step solution with area application.", "Geo Tags: #IsoscelesTriangle #Geometry #TriangleAltitude #PythagoreanTheorem #MathTips"]
