Chapter 3.8, Problem 1PSA

To determine

To prove : The congruency of $\Delta HGJ\text{ and }\Delta KGJ$ .

Explanation of Solution

Given information : The following information has been given

GJ is the altitude to HK

  $\overrightarrow{HG}\cong \overrightarrow{KG}$

Formula used : Any altitude drawn to a line is perpendicular to the original line

Proof : We know that as GJ is the altitude to HK, thus

  $\overrightarrow{GJ}\perp \overrightarrow{HK}$ so now we can say that

  $\angle GJH\cong \angle GJK={90}^0$

Also, GJ is a common chord. Thus, we can say that

  $\overrightarrow{GJ}\cong \overrightarrow{JG}$

And,

  $\overrightarrow{HG}\cong \overrightarrow{KG}$

Now, we can find $\overrightarrow{GH}\cong \overrightarrow{GK}$ by Pythagoras theorem

Thus, we can say that $\Delta HGJ\cong \Delta KGJ$ by SAS property