Let \( a = 5 \) (base), \( c = 13 \) (ladder), solve for \( b \) (height).

["Title: How to Solve for Height ( b ) in a Right Triangle: A Step-by-Step Guide with ( a = 5 ) and ( c = 13 )", "When solving right triangles, one of the most essential relationships to understand is the Pythagorean Theorem:\n[\na^2 + b^2 = c^2\n]\nwhere:\n- ( a ) and ( b ) are the lengths of the legs,\n- ( c ) is the length of the hypotenuse.", "In this article, we’ll explore how to find the unknown leg ( b ) using real-world numbers: let ( a = 5 ) (one leg) and ( c = 13 ) (the hypotenuse).", "---", "### The Problem\nYou’re given:\n- ( a = 5 ) (base leg),\n- ( c = 13 ) (hypotenuse),\n- ( b = ? ) (unknown height).", "We apply the Pythagorean Theorem:\n[\na^2 + b^2 = c^2\n]\nSubstitute the known values:\n[\n5^2 + b^2 = 13^2\n]\n[\n25 + b^2 = 169\n]", "Now isolate ( b^2 ):\n[\nb^2 = 169 - 25\n]\n[\nb^2 = 144\n]", "Take the positive square root (since length cannot be negative):\n[\nb = \sqrt{144} = 12\n]", "---", "### Result\nThe height ( b ) of the triangle is 12 units.", "---", "### Why This Matters\nUnderstanding how to solve for the unknown side in a right triangle is crucial in fields like construction, navigation, physics, and architecture. By applying the Pythagorean Theorem methodically, you can confidently find missing dimensions in any right triangle when two sides are known.", "---", "Key Takeaway:\nUse ( b = \sqrt{c^2 - a^2} ) whenever solving for the missing leg of a right triangle. In our example:\n[\nb = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12\n]", "Master this simple formula, and you’ll solve countless triangle-based problems with ease!", "---", "Tags: Pythagorean Theorem, solve for height, right triangle problems, triangle geometry, solve for ( b ), mathematical formulas, geometry education, practical math examples.\nKeywords: solve for ( b ), right triangle, Pythagorean Theorem, ( a = 5 ), ( c = 13 ), triangle height, solve right triangle, ( b = \sqrt{c^2 - a^2} )"]