Question

the following diagram shows two triangles. the green (upper) triangle has an area of. the purple (lower) triangle has an area of. place the orange triangle (square symbols) directly next to the green triangle so that the two triangles together make a rectangle. the total area of this rectangle is, which is the area of the green triangle.

Explanation:

Step1: Find base and height of green triangle

The green triangle has a base from \(x = 1\) to \(x=5\) (so base \(b_1=4\) inches) and height from \(y = 6\) to \(y = 9\) (so height \(h_1 = 3\) inches).

Step2: Calculate area of green triangle

Using the formula \(A=\frac{1}{2}bh\), for the green triangle \(A_1=\frac{1}{2}\times4\times3 = 6\) square - inches.

Step3: Find base and height of purple triangle

The purple triangle has a base from \(x = 5\) to \(x = 7\) (so base \(b_2 = 2\) inches) and height from \(y = 1\) to \(y = 4\) (so height \(h_2=3\) inches).

Step4: Calculate area of purple triangle

Using the formula \(A=\frac{1}{2}bh\), for the purple triangle \(A_2=\frac{1}{2}\times2\times3=3\) square - inches.

Step5: Analyze orange - green rectangle

If we place the orange triangle next to the green triangle to form a rectangle, the rectangle has length \(l = 4\) inches and width \(w = 3\) inches.

Step6: Calculate area of rectangle

The area of a rectangle is \(A = lw\), so \(A_{rect}=4\times3 = 12\) square - inches. And \(12\) is \(2\) times \(6\).

Answer:

The green (upper) triangle has an area of \(6\) square inches. The purple (lower) triangle has an area of \(3\) square inches. Place the orange triangle (square symbols) directly next to the green triangle so that the two triangles together make a rectangle. The total area of this rectangle is \(12\) square inches, which is \(2\) times the area of the green triangle.