Adding Additional Variables on Menus Page

I'm learning PHP by building a plugin to show or hide menu items based on a user's location. I am a bit stuck at the moment.

In the function below, the text input is being saved to the database, but I'm having trouble saving the radio button data to the database.

I want to the first option to be checked by default.

    function option( $fields, $item_id ) {
    ob_start(); ?>
        <p class="field-visibility description description-wide">
            <label for="edit-menu-item-visibility-<?php echo $item_id; ?>">
                <?php _e('Enter country code(s) separated with commas') ?>:
                <input type="text" 
                        class="widefat code" 
                        id="edit-menu-item-visibility-<?php echo $item_id; ?>" 
                        name="menu-item-visibility[<?php echo $item_id; ?>]" 
                        value="<?php echo esc_html( get_post_meta( $item_id, 'locations', true ) ); ?>" /></br>
                <input type="radio" 
                        id="edit-menu-item-visibility-<?php echo $item_id; ?>" 
                        name="menu-item-show-hide[<?php echo $item_id; ?>]" 
                        value="<?php echo get_post_meta( $item_id, 'hide', true ); ?>" 
                        />Hide from these locations.</br>
                <input type="radio" 
                        id="edit-menu-item-visibility-<?php echo $item_id; ?>"
                        name="menu-item-show-hide[<?php echo $item_id; ?>]" 
                        value="<?php echo get_post_meta( $item_id, 'show', true ); ?>" 
                        />Only show to these locations.</br>
            </label>
        </p>
    <?php
    $fields[] = ob_get_clean();
    return $fields;
    }

Here is the current output:

Ideally, I would use the radio buttons to set a variable called $visible true or false. But I wasn't sure how to accomplish this.

Thanks in advance!