Chapter 6.3, Problem 5PSA

To determine

To find: the slope of the given chord FG and what type of the triangle is $\Delta$ EFG

Answer to Problem 5PSA

Slope of the given chord EF is $\frac{1}{2}$

Slope of the given chord GG is $-2$

Triangle EFG is a right-angle triangle

Explanation of Solution

Given information:

A circle with center O.

One triangle EFG is formed inside the circle

Formula used:

Slope of the line = $\frac{{\text{y}}_2-y_1}{{\text{x}}_2-{\text{x}}_1}$ , where coordinates of the points from which line passes are

  $\left({\text{x}}_1{\text{, y}}_1\right)$ and $\left({\text{x}}_2{\text{, y}}_2\right)$

Angle formed in semi-circle is always right-angle triangle

Product of the slopes of chords is: − 1

Calculation:

End points of the given chord EF are (− 5, 0) and (3, 4)

Slope of the chord EF

   = $\frac{{\text{y}}_2-y_1}{{\text{x}}_2-{\text{x}}_1}$

  $\begin{array}{l}=\frac{4-0}{3+5}=\frac{4}{8}\\ =\frac{1}{2}\end{array}$

Now, end points of FG are (3, 4) and (5, 0) Slope of the chord FG

   = $\frac{{\text{y}}_2-y_1}{{\text{x}}_2-{\text{x}}_1}$

  $\begin{array}{l}=\frac{0-4}{5-3}=-\frac{4}{2}\\ =-2\end{array}$

Product of the slopes of chords EF and FG

  $\begin{array}{l}=\frac{1}{2}\times \left(-2\right)\\ =-1\end{array}$

Since, product of chords EF and FG is (− 1 )

Hence, triangle EFG is a right-angle triangle