Question

right triangle abc is on a coordinate plane. segment ab is on the line \\(y = 2\\) and is 5 units long. point c is on the line \\(x = -2\\). if the area of \\(\delta abc\\) is 12.5 square units, then find a possible y-coordinate of point c.

5
6
7
8

Explanation:

Identify given geometric properties

Using the Right Triangle Coordinates knowledge point

• Segment \(AB\) lies on the horizontal line \(y = 2\) with length \(5\) units.
• Point \(C\) lies on the vertical line \(x = -2\).
• The area of right triangle \(\Delta ABC\) is \(12.5\) square units.

Determine the height of the triangle

Using the Right Triangle Coordinates knowledge point

$$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$

$$ 12.5 = \frac{1}{2} \times 5 \times h \implies h = 5 $$

Analyze right triangle configurations

Using the Right Triangle Coordinates knowledge point

• Since \(AB\) is horizontal on \(y = 2\), the right angle must be at \(A\) or \(B\) for \(\Delta ABC\) to be a right triangle with \(C\) on the vertical line \(x = -2\).
• Thus, one of the vertices \(A\) or \(B\) must have an x-coordinate of \(-2\).
• The vertical distance from the line \(y = 2\) to point \(C\) is the height \(h = 5\).

Calculate possible y-coordinates of C

Using the Right Triangle Coordinates knowledge point

$$ |y_C - 2| = 5 $$

$$ y_C - 2 = 5 \implies y_C = 7 $$

$$ y_C - 2 = -5 \implies y_C = -3 $$

Answer:

• (A) 5
• (B) 6
• (C) 7 (Correct answer)
• (D) 8