Question

3 multiple choice 1 point assuming a dart randomly strikes the interior region of the square, what is the probability that the dart lands in the shaded triangle? 1/8 1/16 3/16 3/8

Explanation:

Step1: Calculate the area of the square

The side - length of the square is \(7\) units (from \(x=-4\) to \(x = 3\) or \(y=-3\) to \(y = 4\)).
The area of a square \(A_{square}=s^{2}\), where \(s = 7\). So \(A_{square}=7\times7 = 49\) (using the formula \(A = s^{2}\)).

Step2: Calculate the area of the shaded triangle

The base of the triangle \(b = 4\) (from \(x=-2\) to \(x = 2\)) and the height \(h=4\) (from \(y = 0\) to \(y = 4\)).
The area of a triangle \(A_{triangle}=\frac{1}{2}\times b\times h\).
Substitute \(b = 4\) and \(h = 4\) into the formula: \(A_{triangle}=\frac{1}{2}\times4\times4=8\).

Step3: Calculate the probability

The probability \(P\) that the dart lands in the shaded region is given by \(P=\frac{A_{triangle}}{A_{square}}\).
Since \(A_{triangle}=8\) and \(A_{square}=49\) is wrong. Wait, re - check the side - length.
Wait, another way: count the number of unit squares.
The square has \(7\times7 = 49\) unit - squares (incorrect approach). Wait, no, assume each grid is \(1\times1\).
The side - length of the square: from \(x=-4\) to \(x = 3\) (distance \(|3-(-4)|=7\)), from \(y=-3\) to \(y = 4\) (distance \(|4 - (-3)|=7\)).
The area of the square \(A_{square}=7\times7=49\) (wrong, because if we consider the grid: assume each small square has side - length \(1\). The number of small squares in the square: \(7\times7 = 49\) (incorrect, because from \(- 4\) to \(3\) (inclusive) in \(x\) - direction: \(8\) units, from \(-3\) to \(4\) (inclusive) in \(y\) - direction: \(8\) units. So \(A_{square}=8\times8=64\)).
The base of the triangle \(b = 4\) (from \(x=-2\) to \(x = 2\)), height \(h = 4\) (from \(y = 0\) to \(y = 4\)).
\(A_{triangle}=\frac{1}{2}\times4\times4 = 8\).
\(P=\frac{A_{triangle}}{A_{square}}=\frac{8}{64}=\frac{1}{8}\).

Answer:

\(\frac{1}{8}\)