Question: A triangle has sides of lengths 10 cm, 10 cm, and 12 cm. What is the length of the shortest altitude?
["A triangle has sides of lengths 10 cm, 10 cm, and 12 cm. What is the length of the shortest altitude?", "Curious readers often turn to precise, visual puzzles like geometric shapes when exploring math, design, or real-world applications. The question “What is the length of the shortest altitude?” in a triangle with sides 10, 10, and 12 cm surfaces naturally—especially in classrooms, DIY projects, or even architectural planning. Users browsed closely, searching for both clarity and confidence in the answer. This query isn’t just about numbers—it’s about unlocking understanding of triangle geometry and how its parts interact.", "### Why This Triangle Matters in the US Context", "Small triangles like this shape everything from garden layouts and fabric patterns to smartphone screen proportions and product design. One side of 12 cm acts as a base for symmetry and balance in visual work, while the equal 10 cm sides invite study of stability and proportion. In a mobile-first environment, where quick, clear knowledge drives intent, users seeking this answer are often solving practical problems—whether designing a quilt, analyzing structural risks, or teaching math. With the growth of visual learning and educational platforms like Discover, questions about triangle altitudes emerge consistently, reflecting a quiet but steady demand for precise, reliable geometry insights.", "### How This Triangle Works: A Clear Breakdown", "Let’s start with the basics: this triangle has two equal sides (10 cm each) and a base of 12 cm—making it an isosceles triangle. Since two sides are equal, it balances stability and asymmetry, a key feature in both nature and construction.", "To find the shortest altitude, begin by calculating the area. Using Heron’s formula simplifies the process: \nFirst, compute the semi-perimeter: \n\( s = \frac{10 + 10 + 12}{2} = 16 \) cm", "Then apply Heron’s formula: \n\( \ ext{Area} = \sqrt{s(s-a)(s-b)(s-c)} \) \n\( = \sqrt{16(16-10)(16-10)(16-12)} = \sqrt{16 \cdot 6 \cdot 6 \cdot 4} = \sqrt{2304} = 48 \) cm²", "With area known, the altitude corresponding to a side is \( \ ext{Altitude} = \frac{2 \ imes \ ext{Area}}{\ ext{Base}} \). \nThe shortest altitude corresponds to the longest base, so comparing 10 cm and 12 cm bases: \n- Altitude to 12 cm side: \( \frac{2 \ imes 48}{12} = 8 \) cm \n- Altitude to 10 cm side: \( \frac"]
