Chapter 8.1, Problem 52HP

To determine

Tocalculate:

The value of $BC$ using given figure.

Answer to Problem 52HP

The value of $BC$ is $\approx 10.0$ .

Explanation of Solution

Given information:

  

Calculation:

Since $△ABC$ is a right triangle, the orthocenter of $△ABC$ is that is at $B$ . Thus $BD=6.4$

The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of this altitude is the geometric mean between the lengths of these two segments.

Geometric mean altitude theorem shows that,

  $BD=\sqrt{\left(AD\right)\left(DC\right)}$

Substitute,

  $\begin{array}{l}6.4=\sqrt{5.3\left(DC\right)}\\ {6.4}^2=5.3\left(DC\right)\\ 40.96=5.3\left(DC\right)\\ 7.73=DC\end{array}$

The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of a leg of this triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

Geometric mean leg theorem shows that,

  $\begin{array}{l}BC=\sqrt{\left(CD\right)\left(CA\right)}\\ =\sqrt{7.73\left(7.73+5.3\right)}\\ =\sqrt{100.722}\\ \approx 10.0\end{array}$

Hence, the value of $BC$ is $\approx 10.0$ .