https://file.aiquickdraw.com/mathbot/file/1726726671535aihmubal.jpg

I have reviewed the image, and I'll now proceed to solve the questions as requested:

1. **Exercise 9 (G(5, -6) and H(x, y)):**
   Midpoint \( M(2, -3) \) is given, so:
   \[
   M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
   \]
   Solving for \( x_2 \) and \( y_2 \), the other endpoint H:
   \[
   H(x, y) = (-1, 0)
   \]

2. **Exercise 10 (G(-3, 7) and H(x, y)):**
   Midpoint \( M(-4, 2) \) is given:
   \[
   H(x, y) = (-5, -3)
   \]

3. **Exercise 11 (G(-2, 9) and H(x, y)):**
   Midpoint \( M(5, 0) \) is given:
   \[
   H(x, y) = (12, -9)
   \]

4. **Exercise 12 (Distance Correction between A(6, 2) and B(1, -4)):**
   Correct distance formula:
   \[
   d = \sqrt{(6 - 1)^2 + (2 - (-4))^2} = \sqrt{5^2 + 6^2} = \sqrt{25 + 36} = \sqrt{61}
   \]
   The error is that the original calculation mistakenly uses squares of the differences directly. The corrected distance remains \( \sqrt{61} \), not 61.

5. **Exercise 13 (Distance Correction):**
   Similarly, there is a miscalculation with the order of operations. The correct distance remains:
   \[
   d = \sqrt{41}
   \]

6. **Exercise 14a (Find the distance around the triangular park):**
   Using the given diagram, the distances between Q, R, and P are directly from the graph:
   - QR = 60 yards
   - PQ = 80 yards
   - PR = 100 yards (this is a right triangle, so these are the side lengths).

   Total distance around the park:
   \[
   60 + 80 + 100 = 240 \text{ yards}
   \]

7. **Exercise 14b (Find QM to the nearest yard):**
   The midpoint M of PR can be calculated, and distance QM is half of PR:
   \[
   QM = \frac{100}{2} = 50 \text{ yards}
   \]

8. **Exercise 14c (Jogging around the park):**
   The total distance jogged is from P to Q to M to R to Q to P, which includes a distance of 50 yards (QM):
   \[
   Total \text{ distance} = PQ + QM + MR + RQ + QP = 80 + 50 + 50 + 60 + 80 = 320 \text{ yards}
   \]
   At 150 yards per minute:
   \[
   \frac{320 \text{ yards}}{150 \text{ yards/min}} \approx 2.13 \text{ minutes}
   \]

If you'd like further details or explanations, feel free to ask.

Here are 5 related questions:
1. What is the formula for finding the distance between two points in the plane?
2. How do you calculate the coordinates of a point given its midpoint and one endpoint?
3. What are the possible errors when calculating the distance between two points?
4. How do you find the perimeter of a triangle given its side lengths?
5. How do you apply the distance formula in real-world problems like the jogger problem?

**Tip:** Always double-check the order of operations when applying formulas like the distance or midpoint formula.

This page covers solutions to problems involving the midpoint formula, distance formula, and geometry of triangles. You'll learn how to calculate missing coordinates, correct errors in distance calculations, and find distances around a park based on a triangular diagram. Suitable for Grade 8-10 math students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation