Question

a. three sidewalks in a schoolyard form a triangle. explain how the expression 1/2(40)(30) + 1/2(40)(20) represents the area inside the three sidewalks.
b. what is the area inside the sidewalks? show your work.

Explanation:

Step1: Recall triangle - area formula

The area formula of a triangle is $A = \frac{1}{2}bh$, where $b$ is the base and $h$ is the height.

Step2: Analyze the given expression for part a

The triangle formed by the sidewalks can be thought of as composed of two non - overlapping right - angled triangles with a common base of length $b = 40$ ft. One right - angled triangle has a height of $h_1=30$ ft and the other has a height of $h_2 = 20$ ft. The area of the first right - angled triangle is $A_1=\frac{1}{2}(40)(30)$ and the area of the second right - angled triangle is $A_2=\frac{1}{2}(40)(20)$. The sum of the areas of these two triangles, $\frac{1}{2}(40)(30)+\frac{1}{2}(40)(20)$, gives the area of the entire triangle formed by the sidewalks.

Step3: Calculate the area for part b

First, calculate $\frac{1}{2}(40)(30)$:
$\frac{1}{2}(40)(30)=20\times30 = 600$
Then, calculate $\frac{1}{2}(40)(20)$:
$\frac{1}{2}(40)(20)=20\times20=400$
Now, find the sum:
$A=\frac{1}{2}(40)(30)+\frac{1}{2}(40)(20)=600 + 400=1000$ square feet.

Answer:

a. The triangle formed by the sidewalks can be split into two right - angled triangles with a common base of 40 ft. The sum of the areas of these two right - angled triangles (using the formula $A=\frac{1}{2}bh$) gives the area of the whole triangle.
b. 1000 square feet