Question

A right triangle is formed by the x-axis, the y-axis, and the line y = -2x + 7.A. Sketch and label a graph of the triangle in the coordinate plane.B. Find the length of the hypotenuse.C. Find the area of the triangle.

Answer 1

The given equation is

`y=-2x+7`

We have to find the axis interceptions to graph this function.

For x = 0, we have

`y=-2(0)+7=0+7=7`

The y-interception is (0,7).

For y = 0, we have

`\begin{gathered} 0=-2x+7 \\ 2x=7 \\ x=(7)/(2) \end{gathered}`

The x-interception is (7/2, 0).

Now, we graph.

Where h is the hypothenuse.

To find the hypothenuse of the right triangle formed, we use the distance formula and both points.

`d=\sqrt[]{(y_2-y_1)^2+(x_2-x_1)^2}`

Replacing the coordinates, we have

`\begin{gathered} d=\sqrt[]{(0-7)^2+(3.5-0)^2}=\sqrt[]{(-7)^2+(3.5)^2} \\ d=\sqrt[]{49+12.25}=\sqrt[]{61.25}\approx7.83 \end{gathered}`

Therefore, the hypotenuse is 7.83 units long, approximately.

At last, the area of the triangle is found with the formula

`A=(1)/(2)b\cdot h`

Where b is the base and h is the height of the triangle. b = 3.5, h = 7.

Replacing these values, we have

`A=(1)/(2)(3.5)\cdot(7)=(24.5)/(2)=12.25u^2`

Therefore, the area of the triangle is 12.25 square units.