Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Connect with professionals ready to provide precise answers to your questions on our comprehensive Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.

Determine the equation of the median from [tex]\( J \)[/tex] for the triangle with vertices [tex]\( J(2,5) \)[/tex], [tex]\( K(4,-1) \)[/tex], and [tex]\( L(-2,-5) \)[/tex]. Write the final answer in slope-intercept form: [tex]\( y = mx + b \)[/tex]. Show your work.

Sagot :

To determine the equation of the median from vertex [tex]\( J \)[/tex] for the triangle with vertices [tex]\( J(2, 5) \)[/tex], [tex]\( K(4, -1) \)[/tex], and [tex]\( L(-2, -5) \)[/tex], we will follow these steps:

1. Find the midpoint of the line segment [tex]\( KL \)[/tex].
2. Calculate the slope of the line that joins vertex [tex]\( J \)[/tex] and the midpoint.
3. Use the point-slope form of the equation of a line to find the equation in slope-intercept form.

### Step 1: Find the midpoint of [tex]\( KL \)[/tex]

The coordinates of the midpoint [tex]\( M \)[/tex] of a line segment between two points [tex]\( K(x_1, y_1) \)[/tex] and [tex]\( L(x_2, y_2) \)[/tex] are given by:

[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]

For points [tex]\( K(4, -1) \)[/tex] and [tex]\( L(-2, -5) \)[/tex]:
[tex]\[ M_x = \frac{4 + (-2)}{2} = \frac{2}{2} = 1 \][/tex]
[tex]\[ M_y = \frac{-1 + (-5)}{2} = \frac{-6}{2} = -3 \][/tex]

So, the midpoint [tex]\( M \)[/tex] is [tex]\((1, -3)\)[/tex].

### Step 2: Calculate the slope of the median

The slope [tex]\( m \)[/tex] of the line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

Using the points [tex]\( J(2, 5) \)[/tex] and [tex]\( M(1, -3) \)[/tex]:
[tex]\[ m = \frac{-3 - 5}{1 - 2} = \frac{-8}{-1} = 8 \][/tex]

The slope of the median line is [tex]\( 8 \)[/tex].

### Step 3: Use the point-slope form to find the equation of the line

The point-slope form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]

Using the slope [tex]\( 8 \)[/tex] and the point [tex]\( J(2, 5) \)[/tex]:
[tex]\[ y - 5 = 8(x - 2) \][/tex]

Rearrange this to the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y - 5 = 8x - 16 \][/tex]
[tex]\[ y = 8x - 16 + 5 \][/tex]
[tex]\[ y = 8x - 11 \][/tex]

### Conclusion

The equation of the median from vertex [tex]\( J \)[/tex] for the triangle with vertices [tex]\( J(2, 5) \)[/tex], [tex]\( K(4, -1) \)[/tex], and [tex]\( L(-2, -5) \)[/tex] is:
[tex]\[ y = 8x - 11 \][/tex]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.