Question

19 mark for review a triangle is formed by the two points shown and a third point (not shown). the area of the triangle is 60 square units. which of the following could be the third point? a (-58,12) b (0,60) c (5,6) d (6,58)

Explanation:

Step1: Identify the two given points

The two points on the graph are \( O(0,0) \) and \( (10,0) \). The distance between these two points (base of the triangle) is \( 10 - 0 = 10 \) units.

Step2: Recall the area formula for a triangle

The area of a triangle is given by \( A=\frac{1}{2}\times base\times height \). We know \( A = 60 \) and \( base = 10 \). Substitute these values into the formula:

$$ 60=\frac{1}{2}\times10\times height $$

Step3: Solve for the height

Simplify the right - hand side: \( \frac{1}{2}\times10 = 5 \), so the equation becomes \( 60 = 5\times height \). Then, divide both sides by 5: \( height=\frac{60}{5}=12 \). The height of the triangle is the vertical distance from the base (the x - axis, since the base is along the x - axis from \( (0,0) \) to \( (10,0) \)) to the third point. So the y - coordinate of the third point should be either \( 12 \) or \( - 12 \) (because height is a distance, but we consider the absolute value of the vertical difference).

Step4: Check the options

• Option A: The point is \( (- 58,12) \). The y - coordinate is 12, which satisfies the height requirement.
• Option B: The point is \( (0,60) \). The y - coordinate is 60, not 12. So this is incorrect.
• Option C: The point is \( (5,6) \). The y - coordinate is 6, not 12. So this is incorrect.
• Option D: The point is \( (6,58) \). The y - coordinate is 58, not 12. So this is incorrect.

Answer:

A. \((-58,12)\)