Median Length Of Equilateral Triangle PQR

```markdown # Length of Median in Equilateral Triangle PQR Hello! Are you trying to find out how to calculate the length of the median in an equilateral triangle? You've come to the right place! In this article, we’ll explain step-by-step how to determine the length of the median in an equilateral triangle, specifically triangle PQR. We will provide a clear, detailed, and correct answer along with a thorough explanation. ## Correct Answer The length of the median in an equilateral triangle PQR is equal to ${\frac{\sqrt{3}}{2} }$ times the side length of the triangle. ## Detailed Explanation Let’s dive into why this is the correct answer. We will start by defining some key concepts and then walk through the process of deriving the formula for the median length in an equilateral triangle. ### Key Concepts * Equilateral Triangle: An equilateral triangle is a triangle in which all three sides are of equal length, and all three angles are equal (each being 60 degrees). * Median: A median of a triangle is a line segment from a vertex (corner point) to the midpoint of the opposite side. * Right Triangle: A right triangle (or right-angled triangle) is a triangle containing one angle that is a right angle (that is, an angle of 90 degrees). * Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed as: ${ a^2 + b^2 = c^2 }$ Where: * a and b are the lengths of the two shorter sides (legs) of the right triangle. * c is the length of the hypotenuse. ### Deriving the Median Length Formula Let’s consider an equilateral triangle PQR, where: * All sides have equal length, which we'll denote as s. * We want to find the length of the median from vertex P to the midpoint M of side QR. Here’s a step-by-step breakdown: 1. Draw the Equilateral Triangle and Median: * Draw an equilateral triangle PQR. * Mark the midpoint of side QR as M. * Draw a line segment from vertex P to M. This is the median, and we want to find its length (PM). 2. Properties of the Median in an Equilateral Triangle: * In an equilateral triangle, the median from any vertex to the midpoint of the opposite side also acts as an altitude. This means that PM is perpendicular to QR. * Therefore, triangle PMQ (or PMR) is a right triangle, with a right angle at M. 3. Determine the Lengths: * Since M is the midpoint of QR, the length of QM (or MR) is half the length of QR. Thus, QM = MR = ${\frac{s}{2}}$. * The length of PQ is s (since PQR is equilateral). 4. Apply the Pythagorean Theorem: * In right triangle PMQ, we have: * PQ (hypotenuse) = s * QM = ${\frac{s}{2}}$ * PM (median) = unknown, let’s call it m * Using the Pythagorean theorem: ${ PQ^2 = PM^2 + QM^2 }$ ${ s^2 = m^2 + (\frac{s}{2})^2 }$ 5. Solve for the Median Length (m): * Substitute the known values into the equation: ${ s^2 = m^2 + \frac{s^2}{4} }$ * To isolate ${m^2}$, subtract ${\frac{s^2}{4}}$ from both sides: ${ m^2 = s^2 - \frac{s^2}{4} }$ * Find a common denominator to combine the terms: ${ m^2 = \frac{4s^2}{4} - \frac{s^2}{4} }$ ${ m^2 = \frac{3s^2}{4} }$ * Take the square root of both sides to solve for m: ${ m = \sqrt{\frac{3s^2}{4}} }$ * Simplify the square root: ${ m = \frac{\sqrt{3} \cdot \sqrt{s^2}}{\sqrt{4}} }$ ${ m = \frac{\sqrt{3}s}{2} }$ Therefore, the length of the median (PM) is ${\frac{\sqrt{3}}{2}s}$. ### Example Calculation Let’s say we have an equilateral triangle PQR with a side length of 10 cm. We can find the length of the median using the formula we derived: ${ m = \frac{\sqrt{3}}{2}s }$ ${ m = \frac{\sqrt{3}}{2} \times 10 }$ ${ m = 5\sqrt{3} }$ ${ m \approx 5 \times 1.732 }$ ${ m \approx 8.66 \text{ cm} }$ So, the length of the median in an equilateral triangle with a side length of 10 cm is approximately 8.66 cm. ### Practical Applications Understanding how to calculate the median of an equilateral triangle is useful in various fields: * Engineering: Calculating structural support and stability. * Architecture: Designing symmetrical structures and spaces. * Geometry: Solving complex geometric problems. * Navigation: Determining distances and directions using triangles. The median's properties and calculations are foundational in many practical scenarios, making this a valuable concept to master. ### Common Mistakes to Avoid * Confusing Median with Altitude: Remember that in an equilateral triangle, the median is the same as the altitude, but this isn't true for all triangles. * Incorrectly Applying the Pythagorean Theorem: Ensure you correctly identify the hypotenuse and the legs of the right triangle. * Forgetting to Simplify: Always simplify the square root to get the final answer in the simplest form. * Using Incorrect Side Length: Double-check that you are using the correct side length in your calculations. By being mindful of these common mistakes, you can improve the accuracy of your calculations. ## Key Takeaways Here are the key points to remember about the length of the median in an equilateral triangle: * The median of an equilateral triangle bisects the opposite side and is also its altitude. * The formula to find the length of the median (${m}$) in an equilateral triangle with side length (${s}$) is: ${ m = \frac{\sqrt{3}}{2}s }$ * The median divides the equilateral triangle into two congruent right triangles. * Understanding this concept is valuable in geometry, engineering, and architecture. I hope this detailed explanation has helped you understand how to find the length of the median in an equilateral triangle. If you have any more questions, feel free to ask!