Pregunta
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Find the area of the triangle defined by the points and .
9
16
18
4.5
QUESTION 3
Which shapes have the same area?
II.
III.
They all have different areas
II and III
I, II and III
I and II

Ask by Fernandez Norton. in the United States
Jan 24,2025

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Question 2:
The area of the triangle is 9.
Question 3:
Shapes I and II have the same area.

Solución

Let’s address each question one by one.

Question 2

Find the area of the triangle defined by the points , and .
Options:
  • 9
  • 16
  • 18
  • 4.5
Solution:
  1. Identify the Base and Height:
    • Points and lie on the same horizontal line ( ), so can serve as the base.
    • Length of base :
    • The height is the vertical distance from point to the line :
  2. Calculate the Area:
Answer: 9

Question 3

Which shapes have the same area?
Given Shapes:
  1. I. Points
  2. II. Points
  3. III. Points
Options:
  • They all have different areas
  • II and III
  • I, II and III
  • I and II
Solution:
  1. Shape I:
    • This is a rectangle with length and width both equal to 3 units.
    • Area:
  2. Shape II:
    • This is a rectangle with length 2 units and height 4.5 units.
    • Area:
  3. Shape III:
    • This is a triangle with a base of 4 units and a height of 2 units.
    • Area:
Conclusion:
  • Shapes I and II both have an area of 9.
  • Shape III has a different area of 4.
Answer: I and II

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Bonus Knowledge

To find the area of the triangle formed by the points , and , we can use the formula for the area of a triangle given by three points , , :
Substituting in the coordinates, we get:
Calculating this gives:
Thus, the area of the triangle is .

Now, for the shapes with the same area:
Calculating the areas, we find:
  1. Shape I is a square with side length 3, leading to an area of .
  2. Shape II is a rectangle with dimensions .
  3. Shape III is a triangle with a base of 4 (from points (4,-3) to (0,-3)) and a height of 2, resulting in an area of .
So, shapes I and II both have the same area of 9, while shape III has an area of 4. Therefore, the correct answer is: I and II.

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