Solve the following problems. Draw the figure for each item.

1. The vertices of a triangle are at (1, 7), (6, -1) and (0, 3). Find the coordinates of the midpoints of the sides.

2. The segment from (-1, 4) to (2, 2) is extended three times its own length. Find the terminal point.

3. The segment joining (2,-4), (2, 3) is divided into two segments, one of which is three fourths as long as the other. Find the point of division. Ans. (6, 0) and (5,-1)

4. The circle with its center at (4,-2) has one end of the diameter at (7, 2). Find the coordinates of the other end of this diameter.

5.A person 6 ft. tall is standing near a street light so that he is 4/10 of the distance form the pole to the tip of his shadows. How high above the ground is the light bulb?

Let's solve each problem step by step and draw the required figures.

### Problem 1: Midpoints of the Sides of a Triangle

**Vertices of the triangle:** (1, 7), (6, -1), and (0, 3).

To find the midpoints of the sides of the triangle, we'll use the midpoint formula:

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

**Calculations:**

1. **Midpoint of side between (1, 7) and (6, -1):**

   \[
   \left( \frac{1 + 6}{2}, \frac{7 + (-1)}{2} \right) = \left( \frac{7}{2}, \frac{6}{2} \right) = (3.5, 3)
   \]

2. **Midpoint of side between (6, -1) and (0, 3):**

   \[
   \left( \frac{6 + 0}{2}, \frac{-1 + 3}{2} \right) = \left( \frac{6}{2}, \frac{2}{2} \right) = (3, 1)
   \]

3. **Midpoint of side between (0, 3) and (1, 7):**

   \[
   \left( \frac{0 + 1}{2}, \frac{3 + 7}{2} \right) = \left( \frac{1}{2}, \frac{10}{2} \right) = (0.5, 5)
   \]

**Midpoints:** (3.5, 3), (3, 1), and (0.5, 5).

### Problem 2: Extended Line Segment

**Segment endpoints:** (-1, 4) and (2, 2).

The length of the line segment is extended three times its own length.

1. **Length of the segment:** Using the distance formula:

   \[
   \text{Length} = \sqrt{(2 - (-1))^2 + (2 - 4)^2} = \sqrt{(2 + 1)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}
   \]

2. **Direction vector from (-1, 4) to (2, 2):**

   \[
   (2 - (-1), 2 - 4) = (3, -2)
   \]

3. **Extending the segment:** Scale the direction vector by 3:

   \[
   (3 \times 3, 3 \times -2) = (9, -6)
   \]

4. **Finding the terminal point:**

   \[
   (2, 2) + (9, -6) = (11, -4)
   \]

**Terminal Point:** (11, -4).

### Problem 3: Dividing the Segment into Two Segments

**Endpoints:** (2, -4) and (2, 3).

**Division ratio:** One segment is \(\frac{3}{4}\) of the other.

1. **Total length of the segment:**

   \[
   \text{Length} = |3 - (-4)| = 7
   \]

2. **Finding the ratio:** Let \(x\) be the length of one segment and \(\frac{3}{4}x\) be the other.

   \[
   x + \frac{3}{4}x = 7 \implies \frac{7}{4}x = 7 \implies x = 4
   \]

   The two segments are 4 and 3 units long.

3. **Point of division:**

   From (2, -4) to (2, 3), the point is 4 units above (2, -4):

   \[
   (-4 + 4) = 0 \implies (2, 0)
   \]

**Point of Division:** (2, 0).

### Problem 4: Finding the Other End of a Diameter of a Circle

**Center of circle:** (4, -2).

**One endpoint of the diameter:** (7, 2).

1. **Midpoint (center) formula:**

   \[
   \text{Center} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
   \]

2. **Setting up equations:**

   \[
   4 = \frac{7 + x}{2}, \quad -2 = \frac{2 + y}{2}
   \]

   Solving:

   \[
   8 = 7 + x \implies x = 1, \quad -4 = 2 + y \implies y = -6
   \]

**Other Endpoint:** (1, -6).

### Problem 5: Height of the Light Bulb

**Person's height:** 6 ft.

**Ratio:** \( \frac{4}{10} \) of the distance from the pole to the tip of the shadow.

Using similar triangles:

1. Let \(x\) be the height of the pole, and \(d\) be the distance of the shadow.

2. From the ratio:

   \[
   6 = \frac{4}{10} \times x
   \]

   Solving for \(x\):

   \[
   6 \times \frac{10}{4} = x \implies x = 15
   \]

**Height of the Light Bulb:** 15 ft.

---

Would you like a detailed explanation or any drawings for these solutions? Here are some related questions:

1. How do you find the centroid of a triangle given its vertices?
2. What is the formula for finding the length of a segment given two points?
3. How do you calculate the slope of a line between two points?
4. How can you determine if three points are collinear?
5. What is the equation of a circle given its center and radius?

**Tip:** When solving geometric problems, drawing figures can help visualize and simplify the problem!

Solve the following problems. Draw the figure for each item.

1. The vertices of a triangle are at (1, 7), (6, -1), and (0, 3). Find the coordinates of the midpoints of the sides.

2. The segment from (-1, 4) to (2, 2) is extended three times its own length. Find the terminal point.

3. The segment joining (2, -4), (2, 3) is divided into two segments, one of which is three-fourths as long as the other. Find the point of division.

4. The circle with its center at (4, -2) has one end of the diameter at (7, 2). Find the coordinates of the other end of this diameter.

5. A person 6 ft. tall is standing near a street light so that he is 4/10 of the distance from the pole to the tip of his shadow. How high above the ground is the light bulb?

Solve coordinate geometry problems involving finding midpoints of triangle sides, extending line segments, dividing line segments in a given ratio, finding endpoints of a diameter of a circle, and using similar triangles to calculate the height of a street light. Suitable for high school students in Grades 9-12.

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