Question

1. make sense and persevere

the base and height of a triangle are each extended 2 cm. what is the area of the shaded region?
how do you know?
triangle diagram with base 8 cm, extended base 2 cm, height x cm, extended height 2 cm, shaded region is a rectangle-like area with width 2 cm and height (x + 2) cm?

Explanation:

Problem 10:

Step 1: Set up the difference

To find how many more items the first plant produces, subtract the second plant's production from the first: $(5x + 11) - (2x - 3)$

Step 2: Simplify the expression

Distribute the negative sign: $5x + 11 - 2x + 3$
Combine like terms: $(5x - 2x) + (11 + 3) = 3x + 14$

Step 1: Area of original triangle

The original triangle has base $8$ cm and height $x$ cm. Area formula: $A_{original} = \frac{1}{2} \times 8 \times x = 4x$

Step 2: Area of new (extended) triangle

New base: $8 + 2 = 10$ cm, new height: $x + 2$ cm. Area: $A_{new} = \frac{1}{2} \times 10 \times (x + 2) = 5(x + 2) = 5x + 10$

Step 3: Area of shaded region

Subtract original area from new area: $(5x + 10) - 4x = x + 10$
(Alternatively, the shaded region can be seen as a trapezoid with bases $x$ and $x + 2$, height $2$: $A = \frac{1}{2} \times 2 \times (x + x + 2) = x + 2$? Wait, correction: Wait, the diagram shows a right triangle extended by 2 cm in base and height. Wait, maybe my initial approach was wrong. Let's re-examine:

Wait, the original triangle has base 8, height x. The extended triangle (after adding 2 cm to base and height) has base 8 + 2 = 10, height x + 2. The shaded region is the area between the two triangles. Wait, no—actually, the shaded region is a trapezoid? Wait, the figure: original triangle (base 8, height x), then a rectangle? No, the diagram shows a right triangle, then a 2 cm extension on base and height, with the shaded area being a vertical strip (2 cm wide) and a horizontal strip? Wait, maybe the correct way:

Wait, the original triangle: area $\frac{1}{2} \times 8 \times x = 4x$.
The new triangle (after extending base by 2 and height by 2) has base 10, height x + 2: area $\frac{1}{2} \times 10 \times (x + 2) = 5x + 10$.
But the shaded region is the difference? Wait, no—actually, the shaded area is the area of the new triangle minus the original triangle? Wait, no, the problem says "the base and height of a triangle are each extended 2 cm". Wait, maybe the shaded region is a trapezoid with bases x and x + 2, height 2? Wait, no, let's look at the diagram: the shaded region is a polygon with vertices at (8,0), (10,0), (10, x + 2), (8, x)? No, maybe the shaded area is a rectangle plus a triangle? Wait, perhaps a better approach:

The shaded region can be calculated as the area of the larger triangle minus the smaller triangle, but wait, no—wait, the original triangle has base 8, height x. The extended part (shaded) is a region with base 2, and the height varies from x to x + 2. Wait, maybe the shaded area is a trapezoid with bases x and x + 2, and height 2. The formula for the area of a trapezoid is $\frac{1}{2} \times (base1 + base2) \times height$. So here, base1 = x, base2 = x + 2, height = 2. Then:

$A = \frac{1}{2} \times 2 \times (x + x + 2) = (2x + 2) = x + 2$? Wait, that contradicts the earlier. Wait, maybe I misread the diagram. Let's check again:

The original triangle: right triangle, base 8, height x (vertical side). Then, we extend the base by 2 cm (so total base 10) and the height by 2 cm (so total height x + 2). The shaded region is the area between the original triangle and the new triangle? No, the shaded region is a vertical strip (2 cm wide) on the right and a horizontal strip (2 cm tall) on the top? Wait, no, the diagram shows a right triangle, then a 2 cm extension to the base (making base 10) and a 2 cm extension to the height (making height x + 2). The shaded area is the area of the new (larger) triangle minus the original triangle? Wait, no—actually, the shaded region is a trapezoid with bases x and x + 2, and height 2 (the horizontal extension), plus a rectangle? No, maybe the correct way is:

Wait, the original triangle: area $\frac{1}{2} \times 8 \times x = 4x$.
The new triangle (after extending base by 2 and height by 2) has base 10, height x + 2: area $\frac{1}{2} \times 10 \…

Answer:

$3x + 14$

Problem 12: