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Difficulty: MediumGeometric Figures on the Coordinate Plane

In the standard (x,y)(x, y) coordinate plane, a triangle has vertices at A(−3,−2)A(-3, -2), B(5,−2)B(5, -2), and C(x,y)C(x, y). If the area of the triangle is 2424 square units, which of the following could be the coordinates of vertex CC?

  1. A
    (2,1)(2, 1)
  2. (2,4)(2, 4)Answer
  3. C
    (2,6)(2, 6)
  4. D
    (2,8)(2, 8)
  5. E
    (2,10)(2, 10)

Answer

(2,4)(2, 4)
The correct answer is the coordinate (2,4)(2, 4). The base of the triangle is the segment ABAB, which is horizontal since both vertices lie on the line y=−2y = -2. The length of this base is 5−(−3)=85 - (-3) = 8 units. Using the formula for the area of a triangle, Area=12×base×height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}, we have 24=12×8×height24 = \frac{1}{2} \times 8 \times \text{height}, which simplifies to 24=4×height24 = 4 \times \text{height}, giving a height of 66 units. Since the base lies along y=−2y = -2, the yy-coordinate of the third vertex must be 66 units away from −2-2 (either at −2+6=4-2 + 6 = 4 or −2−6=−8-2 - 6 = -8). The point (2,4)(2, 4) satisfies this requirement.

Step-by-Step Solution

1
Calculate the length of the base of the triangle.
The base segment ABAB is horizontal because both A(−3,−2)A(-3, -2) and B(5,−2)B(5, -2) have the same yy-coordinate of −2-2. The length of the base is the difference in their xx-coordinates: 5−(−3)=85 - (-3) = 8 units.
To use the area formula of a triangle, we first need to determine the length of one of its sides to act as the base.
2
Use the area formula of a triangle to find its height.
The formula for the area of a triangle is Area=12×base×height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}. Substituting the given area of 2424 and the base length of 88: 24=12×8×height⇒24=4×height⇒height=624 = \frac{1}{2} \times 8 \times \text{height} \Rightarrow 24 = 4 \times \text{height} \Rightarrow \text{height} = 6 units.
Knowing the area and the base allows us to find the vertical distance (height) from the base to the third vertex.
3
Determine the possible yy-coordinates of vertex CC.
Since the base lies on the horizontal line y=−2y = -2, the yy-coordinate of vertex CC must be exactly 66 units away from −2-2 vertically. This means y=−2+6=4y = -2 + 6 = 4 or y=−2−6=−8y = -2 - 6 = -8. Thus, CC can be any point with a yy-coordinate of 44 or −8-8.
The height represents the vertical distance from the horizontal base line to the third vertex.
4
Match the calculated yy-coordinates with the given options.
Among the options, only the coordinate (2,4)(2, 4) has a yy-coordinate of 44.
We identify the option that provides a valid set of coordinates for vertex CC.

Key Concept

Calculating coordinates of a geometric figure's vertex using the area formula and coordinate distances.
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Topics

Geometric Figures on the Coordinate Plane

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