5**Question:** An isosceles triangle has a base of 10 cm and legs of 13 cm. If the base is increased by 2 cm, what is the change in the area of the triangle?

["---", "5 Key Questions: Understanding How Changing the Base Affects the Area of an Isosceles Triangle", "When studying triangles in geometry, understanding how changes in dimensions affect area is essential. One classic example involves isosceles triangles — shapes with two equal legs and a base. Consider this common scenario: An isosceles triangle with a base of 10 cm and equal legs of 13 cm. If the base is increased by 2 cm to 12 cm, what is the change in the area? Let’s explore this step by step.", "### Why This Problem Matters", "Calculating the area of triangles helps in real-world applications—from architecture and design to physics and navigation. This problem specifically tests your ability to:", "- Use the correct area formula for an isosceles triangle.\n- Work with height determined via the Pythagorean theorem.\n- Compare areas before and after a change.", "It also highlights a key geometric concept: how altering one side while maintaining equality in a triangle alters its height and, consequently, its area.", "### Step 1: Find the Original Height", "Given:\n- Base = 10 cm → half-base = 5 cm\n- Legs = 13 cm", "Drop a perpendicular from the apex (top vertex) to the midpoint of the base. This splits the isosceles triangle into two congruent right triangles, each with:\n- Hypotenuse = 13 cm\n- Base leg = 5 cm", "Using the Pythagorean theorem to find the height (h):", "[\nh = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12~\ ext{cm}\n]", "### Step 2: Calculate the Original Area", "Use the triangle area formula:", "[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height} = \frac{1}{2} \ imes 10 \ imes 12 = 60~\ ext{cm}^2\n]", "### Step 3: Find the New Height After Base Increase", "Now the base increases to 12 cm → new half-base = 6 cm. The legs remain 13 cm.", "Repeat the Pythagorean approach:", "[\nh' = \sqrt{13^2 - 6^2} = \sqrt{169 - 36} = \sqrt{133} \approx 11.53~\ ext{cm}\n]", "(We keep it exact for precision: $ h' = \sqrt{133} $ cm)", "### Step 4: Compute the New Area", "[\n\ ext{New Area} = \frac{1}{2} \ imes 12 \ imes \sqrt{133} = 6\sqrt{133}~\ ext{cm}^2\n\approx 69.18~\ ext{cm}^2\n]", "### Step 5: Determine the Change in Area", "[\n\Delta \ ext{Area} = \ ext{New Area} - \ ext{Original Area} = 6\sqrt{133} - 60\n]", "Approximately:\n[\n69.18 - 60 = 9.18~\ ext{cm}^2\n]", "But more precisely:", "[\n\Delta \ ext{Area} = 6\sqrt{133} - 60\n]", "### Final Thoughts", "Increasing the base from 10 cm to 12 cm, while keeping the legs fixed, lengthens the height slightly at first but the base increases more — resulting in a net gain in area. The exact change is $ 6\sqrt{133} - 60 $ square centimeters.", "This problem illustrates how geometry evolves with dimension changes and reinforces the importance of base-height relationships in area calculations. Ready to solve the next triangle puzzle?", "---", "Keywords for SEO: isosceles triangle area change, triangle height calculation, change in triangle area, Pythagorean theorem in triangles, base increase effect, geometry problem solving", "---", "Whether you’re a student, teacher, or math enthusiast, mastering such questions improves spatial reasoning and algebraic thinking — essential tools for geometric mastery.", "---", "Help your students visualize these changes with graphing tools or hands-on models. Understanding area evolution helps build deeper geometry intuition.", "---", "Keywords Repeats (for SEO effectiveness):\nisosceles triangle area change, triangle area with base increase, right triangle method, change in triangle area, Pythagorean theorem triangle, geometry area problem", "---", "Stay curious — every triangle tells a story through numbers.\n---"]