Chapter 15, Problem 32CR

To determine

To find: the coordinates of the point C.

Answer to Problem 32CR

  $C\left(-2,5\right)$ .

Explanation of Solution

Given:

CD is the altitudes of the hypotenuse of the $\Delta ABC$ .

The coordinates of the points $A,B\text{and}D$ in the below triangle.

  

Calculation:

According to the question, The coordinates of the points $A,B\text{and}D$ in the triangle.

Now, find the length of the sides of the triangle by using the distance formula. Then,

  $\begin{array}{l}AD=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(-6-\left(-2\right)\right)}^2+{\left(-1-\left(-1\right)\right)}^2}\\ =\sqrt{{\left(-6+2\right)}^2+{\left(-1+1\right)}^2}\\ =\sqrt{16+0}=4\end{array}$

  $\begin{array}{l}BD=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(-2-7\right)}^2+{\left(-1-\left(-1\right)\right)}^2}\\ =\sqrt{{\left(-9\right)}^2+{\left(-1+1\right)}^2}\\ =\sqrt{81+0}=9\end{array}$

Altitude is the mean proportion to the segment of hypotenuse. Thus,

  $\begin{array}{l}\frac{CD}{AD}=\frac{DB}{CD}\\ \Rightarrow C{D}^2=AD\cdot DB\\ C{D}^2=4\cdot 9=36\\ CD=\sqrt{36}=6\end{array}$

Now, point C is unit above point D, so its coordinate will be $-1+6=-5$ .

Thus, the coordinates of the point C is $C\left(-2,5\right)$ .