Question

a rectangle has the dimensions shown below.
select the triangle(s) whose area is half the area of the rectangle.
(rectangle with length 6 and height 4; triangles: first with base 3, height 4; second with base 6, height 2; third with base 6, height 4; fourth with base 6, height 4 (checked); fifth with base 3, height 2)

Explanation:

Step1: Calculate rectangle area

The area of a rectangle is given by the formula \( A_{rectangle} = length \times width \). For the given rectangle, length = 6 and width = 4, so \( A_{rectangle} = 6 \times 4 = 24 \). Half of this area is \( \frac{24}{2} = 12 \).

Step2: Calculate area of each triangle

• First triangle (base 3, height 4): The area of a triangle is \( A_{triangle} = \frac{1}{2} \times base \times height \). So \( A = \frac{1}{2} \times 3 \times 4 = 6

eq 12 \).

• Second triangle (base 6, height 2): \( A = \frac{1}{2} \times 6 \times 2 = 6

eq 12 \).

• Third triangle (base 6, height 4): \( A = \frac{1}{2} \times 6 \times 4 = 12 \), which matches half the rectangle's area.
• Fourth triangle (base 6, height 4): Same as the third, \( A = \frac{1}{2} \times 6 \times 4 = 12 \), matches.
• Fifth triangle (base 3, height 2): \( A = \frac{1}{2} \times 3 \times 2 = 3

eq 12 \).

Answer:

The triangles with base 6 and height 4 (the third and fourth triangles in the list, or the ones with base 6 and height 4) have an area half of the rectangle's area. Specifically, the third triangle (with base 6, height 4), the fourth triangle (with base 6, height 4) are the correct ones. If we consider the checkboxes, the third (the one with base 6, height 4, middle - lower before the last) and the fourth (the one with base 6, height 4, the yellow - boxed one) should be selected. But based on calculation, the triangles with area 12 are the ones with base 6 and height 4. So the correct triangles are the third (base 6, height 4), fourth (base 6, height 4) from the top (or the ones with base 6 and height 4). In terms of the checkboxes, the third (the one with base 6, height 4, the un - checked one above the yellow - boxed) and the fourth (yellow - boxed) should be checked. Wait, the yellow - boxed one has base 6 and height 4, area \( \frac{1}{2}×6×4 = 12 \), the third one (the one with base 6, height 4, above the yellow - boxed) also has base 6 and height 4, area 12. The second triangle: base 6, height 2, area \( \frac{1}{2}×6×2 = 6 \), first: \( \frac{1}{2}×3×4 = 6 \), fifth: \( \frac{1}{2}×3×2 = 3 \). So the correct triangles are the third (the one with base 6, height 4, the one with base 6 and height 4, third from top) and the fourth (yellow - boxed, fourth from top) triangles. So the answer is the third (base 6, height 4) and fourth (base 6, height 4) triangles (the ones with base 6 and height 4). If we list their positions: the third triangle (from top, base 6, height 4), the fourth triangle (from top, base 6, height 4, yellow - boxed).