Question

1. here are three triangles drawn inside three identical rectangles. mariam states that all three triangles have the same area. is mariam correct? explain your thinking?

Explanation:

Step1: Recall rectangle and triangle area formulas

The area of a rectangle is given by \( A_{rectangle} = length \times width \) (let's denote length as \( l \) and width as \( w \), so \( A_{rectangle}=l\times w \)). The area of a triangle is \( A_{triangle}=\frac{1}{2}\times base\times height \).

Step2: Analyze each triangle's base and height

For each triangle inside the identical rectangles:

• In the first rectangle, the triangle has a base equal to the length of the rectangle (\( l \)) and height equal to the width of the rectangle (\( w \)). So its area is \( \frac{1}{2}\times l\times w \).
• In the second rectangle, the triangle has a base equal to the length of the rectangle (\( l \)) and height equal to the width of the rectangle (\( w \)) (because the height is the vertical side of the rectangle). So its area is \( \frac{1}{2}\times l\times w \).
• In the third rectangle, the triangle has a base equal to the length of the rectangle (\( l \)) and height equal to the width of the rectangle (\( w \)) (the height is the vertical distance, same as the rectangle's width). So its area is \( \frac{1}{2}\times l\times w \).

Step3: Compare the areas

Since all three rectangles are identical, \( l \) and \( w \) are the same for each. Thus, each triangle has an area of \( \frac{1}{2}\times l\times w \), which means their areas are equal.

Answer:

Mariam is correct. Because each triangle has a base equal to the rectangle's length and height equal to the rectangle's width, and the area of a triangle is \(\frac{1}{2}\times base\times height\). Since the rectangles are identical (same length and width), each triangle's area is \(\frac{1}{2}\times length\times width\), so all three triangles have the same area.