Question

select the correct answer.
what is the area of the triangle in the diagram?
a. $\frac{1}{2}(x_{1}-y_{1})(x_{2}-y_{1})$
b. $\frac{1}{2}y_{1}(x_{2}-x_{1})$
c. $\frac{1}{2}(x_{2}-x_{1})$
d. $\frac{1}{2\sqrt{(x_{2}^{2}-y_{1}^{2})(x_{1}^{2}-y_{1}^{2})}}$

Explanation:

Step1: Find the base of the triangle

The base of the triangle is the distance between \((x_1,y_1)\) and \((x_2,y_1)\). Since the \(y -\)coordinates are the same, the base \(b=\vert x_2 - x_1\vert\).

Step2: Find the height of the triangle

The height of the triangle is the vertical distance from the line \(y = y_1\) to the origin \((0,0)\). So the height \(h=\vert y_1\vert\).

Step3: Use the area formula for a triangle

The area formula for a triangle is \(A=\frac{1}{2}\times base\times height\). Substituting \(b = x_2 - x_1\) (assuming \(x_2>x_1\)) and \(h = y_1\) (assuming \(y_1>0\)), we get \(A=\frac{1}{2}y_1(x_2 - x_1)\).

Answer:

B. \(\frac{1}{2}y_1(x_2 - x_1)\)