A point M on a segment with endpoints X (1, −2) and Y (10, 3) partitions the segment in a 5:1 ratio. Find M. You must show all work to receive credit.

Question

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Question 4 (Essay Worth 10 points)
(04.03 MC)

Find the area of the following shape. You must show all work to receive credit.

shape with vertices located at negative 4 comma 0, 0 comma 4, 2 comma 2, negative 1 comma 2, and negative 1 comma 0

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FDK71.12

Answer 1

To solve the problems presented, let's break them down step by step.

Part 1: Finding Point M

Given endpoints \(X(1, -2)\) and \(Y(10, 3)\) that are to be divided by point \(M\) in a ratio of \(5:1\):

1. Determine the total parts in the ratio: \[ 5 + 1 = 6 \text{ parts} \]

2. Calculate the coordinates of point \(M\) using the section formula: The section formula states that the coordinates of point \(M\) dividing segment \(XY\) in the ratio \(m:n\) are given by: \[ M\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \] where \( (x_1, y_1) \) are the coordinates of \(X\) and \( (x_2, y_2) \) are the coordinates of \(Y\). Here, \(m = 5\) and \(n = 1\).

3. Substitute the coordinates: \[ x_1 = 1, \quad y_1 = -2, \quad x_2 = 10, \quad y_2 = 3 \] Replace the values in the section formula: \[ M_x = \frac{5 \cdot 10 + 1 \cdot 1}{6} = \frac{50 + 1}{6} = \frac{51}{6} = 8.5 \] \[ M_y = \frac{5 \cdot 3 + 1 \cdot (-2)}{6} = \frac{15 - 2}{6} = \frac{13}{6} \approx 2.1667 \]

4. The coordinates of point M are: \[ M\left(8.5, \frac{13}{6}\right) \]

Part 2: Finding the Area of the Shape

The shape is defined by vertices located at \((-4, 0)\), \((0, 4)\), \((2, 2)\), \((-1, 2)\), and \((-1, 0)\). To find the area, we can use the Shoelace Theorem.

1. List the vertices in order (cyclically): \[ (-4, 0), (0, 4), (2, 2), (-1, 2), (-1, 0) \]

2. Use the Shoelace theorem: The area \(A\) is given by: \[ A = \frac{1}{2} \left| \sum_{i=1}^{n} (x_iy_{i+1} - y_ix_{i+1}) \right| \]

   Where \((x_{n+1}, y_{n+1}) = (x_1, y_1)\) to close the polygon.

3. Calculate the coordinates and their products: \[ (-4, 0) \to 0 \cdot 4 = 0 \quad \text{and} \quad -4 \cdot 0 = 0 \] \[ (0, 4) \to 4 \cdot 2 = 8 \quad \text{and} \quad 0 \cdot 2 = 0 \] \[ (2, 2) \to 2 \cdot 2 = 4 \quad \text{and} \quad 2 \cdot -1 = -2 \] \[ (-1, 2) \to 2 \cdot 0 = 0 \quad \text{and} \quad -1 \cdot 0 = 0 \] \[ (-1, 0) \to 0 \cdot -4 = 0 \quad \text{and} \quad -1 \cdot 0 = 0 \]

Putting this all together:

• Sum of \(x_iy_{i+1}\): \[ 0 + 8 + 4 + 0 + 0 = 12 \]
• Sum of \(y_ix_{i+1}\): \[ 0 + 0 - 2 + 0 + 0 = -2 \]

4. Calculate the area \(A\): \[ A = \frac{1}{2} \left| 12 - (-2) \right| = \frac{1}{2} \left| 12 + 2 \right| = \frac{1}{2} \cdot 14 = 7 \]

Thus, the area of the shape is \(7\) square units.

Final Answers:

• Coordinates of point M: \(M\left(8.5, \frac{13}{6}\right)\)
• Area of the shape: \(7\) square units.