Question

Use the image to answer the question.

A triangle is graphed by joining three plotted points on a coordinate plane. The x-axis ranges from negative 6 to 10 in increments of 2 and the y-axis ranges from negative 4 to 10 in increments of 2.

Amaury draws an acute triangle on the coordinate grid. He finds the area of the triangle by drawing a rectangle that circumscribes the triangle and forming right triangles. Show the steps Amaury must take to find the area of the triangle. What is the area of the triangle in square units?

(4 points)

A(-3,5) B(3,7) C(2,3)

Answer 1

Step 1 — circumscribing rectangle:
- Min x = −3, max x = 3 ⇒ width = 3 − (−3) = 6.
- Min y = 3, max y = 7 ⇒ height = 7 − 3 = 4.
- Rectangle area = 6 × 4 = 24.

Step 2 — identify the three right triangles in the rectangle that lie outside triangle ABC:
- Top triangle with vertices (−3,7), (3,7), (−3,5). Legs = 6 (horizontal) and 2 (vertical) ⇒ area = 1/2·6·2 = 6.
- Right-bottom triangle with vertices (3,7), (3,3), (2,3). Legs = 4 (vertical) and 1 (horizontal) ⇒ area = 1/2·4·1 = 2.
- Left-bottom triangle with vertices (−3,3), (−3,5), (2,3). Legs = 2 (vertical) and 5 (horizontal) ⇒ area = 1/2·2·5 = 5.

Step 3 — subtract the three triangle areas from the rectangle:
- Total area removed = 6 + 2 + 5 = 13.
- Area of triangle ABC = 24 − 13 = 11 square units.

Answer: 11 square units.