/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1322 If a vertex of a triangle is \((... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 1322

If a vertex of a triangle is \((1,1)\) and the mid-points of two sides through this vertex are \((-1,2)\) and \((3,2)\), then centroid of the triangle is (a) \([(1 / 3),(7 / 3)]\) (b) \([1,(7 / 3)]\) (c) \([-(1 / 3),(7 / 3)]\) (d) \([-1,(7 / 3)]\)

Short Answer

Expert verified
The centroid of the triangle is (1, 7/3), so the short answer is option (b) $[1,(\frac{7}{3})]$.

Step by step solution

01

Identify given information and notations

Given, a vertex of a triangle is A(1, 1) and the midpoints of two sides through this vertex are M1(-1, 2) and M2(3, 2). Let the other vertices of the triangle be B and C.
02

Find the coordinates of point B

Since M1 is the midpoint of AB, we can use the midpoint formula to find the coordinates of B: Midpoint formula: \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\) Substitute the coordinates of point A(1,1) and M1(-1,2), we have: \((-1, 2) = \left( \frac{1 + x_B}{2}, \frac{1 + y_B}{2} \right)\) Now, solve for x_B and y_B: \(x_B = 2 \times (-1) - 1 = -3\) \(y_B = 2 \times 2 - 1 = 3\) So, the coordinates of point B are (-3, 3).
03

Find the coordinates of point C

Similarly, since M2 is the midpoint of AC, we can use the midpoint formula to find the coordinates of C: Substitute the coordinates of point A(1,1) and M2(3,2), we have: \((3, 2) = \left( \frac{1 + x_C}{2}, \frac{1 + y_C}{2} \right)\) Now, solve for x_C and y_C: \(x_C = 2 \times 3 - 1 = 5\) \(y_C = 2 \times 2 - 1 = 3\) So, the coordinates of point C are (5, 3).
04

Calculate the centroid coordinates

The centroid of a triangle can be found by taking the average of its vertices' x-coordinates and y-coordinates: Centroid: \(\left( \frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3} \right)\) Substitute the coordinates of points A(1,1), B(-3,3), and C(5,3), we have: Centroid: \(\left( \frac{1 - 3 + 5}{3}, \frac{1 + 3 + 3}{3} \right)\) Simplifying the coordinates, we get: Centroid: \(\left( \frac{3}{3}, \frac{7}{3} \right) = (1, \frac{7}{3})\) The centroid of the triangle is (1, 7/3), so our answer is option (b).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Midpoint Formula
To understand the solution to a triangle centroid problem, it's essential to grasp the concept of the midpoint formula. The midpoint formula helps us find the middle point of a line segment in geometry. If you have two endpoints of a line segment, say
  • Point 1: \(x_1, y_1\)
  • Point 2: \(x_2, y_2\)
The formula to calculate the midpoint is:\[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]This formula averages the respective coordinates of the endpoints. Using this will give you the exact center point on the line connecting the two original points, which is especially useful when partitioning geometric shapes or determining triangles' vertices.
Coordinates of Triangle Vertices
Finding the coordinates of triangle vertices involves using known points and the midpoint formula. In problems where you're given a vertex and the midpoints of sides passing through this vertex, like
  • Vertex A: \(1, 1\)
  • Midpoints: \(M_1 = (-1, 2)\) and \(M_2 = (3, 2)\)
you can find the unknown points. Here, we calculate the unknown vertices \(B\) and \(C\) using:
  • For midpoint \(M_1\) (point A to B): Solve \(\left( \frac{1 + x_B}{2}, \frac{1 + y_B}{2} \right) = (-1, 2)\)
  • For midpoint \(M_2\) (point A to C): Solve \(\left( \frac{1 + x_C}{2}, \frac{1 + y_C}{2} \right) = (3, 2)\)
By solving these equations, you find that B and C have coordinates \((-3, 3)\) and \(5, 3\), respectively. Understanding these steps helps in visualizing and structuring the geometrical setup of the triangle.
Geometry Problems
Geometry problems often involve understanding spatial relationships between points and lines. In the case of triangles, aspects like vertex positioning, side calculations, and centroid determination are crucial.
  • First, you need to visualize the triangle given specific points and midpoints.
  • Next, solving for unknown vertices and ensuring each calculation checks out with known geometrical properties.
Once vertices are known, calculations such as finding centroids becomes straightforward. The centroid divides each median into a ratio of 2:1, and it acts as the triangle's center of balance. It is computed as the average of the x-coordinates and y-coordinates of the vertices, reflecting the triangle's symmetry and equilibrium.
JEE Mathematics
JEE Mathematics requires a strong foundation in geometric principles to answer questions effectively.
  • Your ability to apply basic formulas like the midpoint and centroid determination in problem-solving is tested.
  • Conceptual clarity allows students to approach and solve complex triangle problems efficiently.
Students preparing for JEE should practice problems involving triangle properties and calculations to develop a keen insight into analyzing geometrical figures. Mastery of these methods ensures an edge in exams, as many questions demand quick thinking and precise calculations of geometric constructs like centroids.

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