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. $\frac{1}{2\sqrt{(x_2^2 - x_1^2)(x_2^2 - x_1^2)}}$

Explanation:

Step1: Recall the formula for the area of a triangle

The formula for the area of a triangle is \(A=\frac{1}{2}\times base\times height\).

Step2: Identify the base and height of the triangle

In the given diagram, the base of the triangle is the distance between the points \((x_1,y_1)\) and \((x_2,y_1)\). Using the distance formula for two points \((x_1,y_1)\) and \((x_2,y_1)\) (since \(y - \)coordinates are the same), the base \(b=\vert x_2 - x_1\vert\). The height of the triangle is the perpendicular distance from the line \(y = y_1\) to the origin \((0,0)\), which is \(h=\vert y_1\vert\) (since the line \(y = y_1\) is horizontal and the vertical distance from \((0,0)\) to \(y = y_1\) is \(y_1\)).

Step3: Calculate the area of the triangle

Substitute the base \(b=(x_2 - x_1)\) (assuming \(x_2>x_1\)) and height \(h = y_1\) into the area formula \(A=\frac{1}{2}\times base\times height\). So \(A=\frac{1}{2}y_1(x_2 - x_1)\).

Answer:

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