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.

Explanation:

Identify the given geometric properties

We are given a right triangle \(ABC\) on a coordinate plane:

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

Relate the area to the base and height

Since segment \(AB\) is horizontal along \(y = 2\), we can treat \(AB\) as the base of \(\triangle ABC\).

• Base \(b = AB = 5\).
• The height \(h\) is the perpendicular distance from point \(C\) to the line containing \(AB\) (which is \(y = 2\)).

Using the area formula:

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

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

$$ 12.5 = 2.5 \times h \implies h = 5 $$

Determine the possible y-coordinates of point C

The height \(h = 5\) represents the vertical distance from point \(C\) to the horizontal line \(y = 2\).
Let the coordinates of point \(C\) be \((-2, y_C)\) since \(C\) lies on the line \(x = -2\).
The vertical distance is given by:

$$ |y_C - 2| = 5 $$

This yields two possible cases:

1. \(y_C - 2 = 5 \implies y_C = 7\)
2. \(y_C - 2 = -5 \implies y_C = -3\)

Verify the right triangle condition

We must ensure that \(\triangle ABC\) is a right triangle.

• Since \(AB\) is horizontal (slope is \(0\)), any vertical segment is perpendicular to \(AB\).
• Point \(C\) lies on the vertical line \(x = -2\).
• If one of the vertices \(A\) or \(B\) has an x-coordinate of \(-2\), then the side \(AC\) or \(BC\) will be vertical, forming a right angle at that vertex (since horizontal and vertical lines are perpendicular).
• Since \(AB\) has length \(5\) along \(y = 2\), we can choose coordinates for \(A\) and \(B\) such that one of them lies on \(x = -2\). For example, let \(A = (-2, 2)\) and \(B = (3, 2)\). Then \(AB\) is horizontal with length \(5\).
• With \(C = (-2, 7)\) or \(C = (-2, -3)\), the segment \(AC\) lies on the vertical line \(x = -2\).
• Thus, \(AC \perp AB\), making \(\triangle ABC\) a right triangle with a right angle at \(A\).
• Therefore, both \(y = 7\) and \(y = -3\) are valid possible y-coordinates.

Answer:

A possible y-coordinate of point C is 7 (or -3).