4.1 Q-5

Question Statement

Given the midpoints of the sides of a triangle at coordinates $(1,-1)$, $(-4,-3)$, and $(-1,1)$, find the coordinates of the vertices of the triangle.

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Background and Explanation

This problem involves finding the coordinates of the vertices of a triangle when given the midpoints of its sides. To solve it, we will use the property that the midpoint of a line segment joining two points is the average of the coordinates of those points. The midpoint formula is:

$$\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$

We will use this formula to set up equations relating the coordinates of the triangle’s vertices and their midpoints.

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Solution

We are given that the midpoints of the sides of the triangle are:

• $D(1,-1)$
• $E(-4,-3)$
• $F(-1,1)$

Let the vertices of the triangle be:

• $A(a, b)$
• $B(c, d)$
• $C(e, f)$

Using the midpoint formula, we can write the following system of equations based on the midpoints of the sides:

• For midpoint $D$ of side $BC$:
  $\frac{c + e}{2} = 1 \quad \text{and} \quad \frac{d + f}{2} = -1$
  This simplifies to: c + e = 2 \tag{1}
  d + f = -2 \tag{2}

• For midpoint $E$ of side $AC$:
  $\frac{a + e}{2} = -4 \quad \text{and} \quad \frac{b + f}{2} = -3$
  This simplifies to: a + e = -8 \tag{3}
  b + f = -6 \tag{4}

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Key Formulas or Methods Used

• Midpoint Formula:
  $\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

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Summary of Steps

1. Set up equations based on the midpoint formula for each side of the triangle.
2. Solve the system of equations to find the coordinates of the vertices.
3. Use substitution and elimination to solve for the unknowns.

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Now, let’s solve the system of equations step-by-step:

1. From equation (1):
   $c + e = 2$
   From equation (3):
   $a + e = -8$

Adding these two equations gives: a + c + 2e = -6 \tag{5}

2. From equation (2):
   $d + f = -2$
   From equation (4):
   $b + f = -6$

Adding these two equations gives: b + d + 2f = -8 \tag{6}

3. Using equation (5) and simplifying: $a + c + 2e = -6 \quad \text{(substitute $c + e = 2$)}$
   $-2 + 2e = -6$
   $2e = -4$
   $e = -2$

4. Substitute $e = -2$ into equation (3): $a + (-2) = -8$
   $a = -6$

5. Now substitute $e = -2$ into equation (4): $b + f = -6$
   Substitute $f = -5$: $b - 5 = -6$
   $b = -1$

6. Finally, substitute $f = -5$ into equation (2): $d + (-5) = -2$
   $d = 3$

Thus, the coordinates of the vertices are:

• $A(-6, -1)$
• $B(4, 3)$
• $C(-2, -5)$

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