Okay -- output shows two things.
1. Only one Linux install is currently on there -- don't know which one.
2. That Linux was installed in BIOS/Legacy mode; but needed to be installed in UEFI-mode.
Since Windows is using UEFI-mode and GPT partitions, you must install Linux in the same mode and the version you install must be a 64-bit version. During power-up of computer you'll need to play around with UEFI Settings/Startup menu and/or the Boot Menu -- both of which are activated by hitting different special keys upon power-up. (You'll have to consult computer's user manual for info on which keys do that.)
It's possible that you might be able to convert the Linux install that you have on there now to UEFI-mode; so try that first before resorting to a re-installation.
Before proceeding on any of this -- repair or re-install attempt -- be sure to have backups of anything important on the drive and make Win8 recovery disks if you haven't already done so.
To attempt conversion, use the live DVD of the Linux that you know is installed currently (probably which ever one you installed last) and boot it in UEFI/EFI-mode. Confirm that it is booted in UEFI-mode by entering this in a terminal and look for "efi" as one of the things listed:
If "efi" is not listed, then you're still booted in Legacy-mode. Change UEFI boot settings and try again.
Once UEFI-boot confirmed, install Boot-Repair utility with intructions here, but DO NOT proceed with those instructions once Boot-Repair is started up: https://help.ubuntu.com/community/Boot-Repair
. When Boot-Repair starts, use these instructions for converting the current install to UEFI-mode: https://help.ubuntu.com/community/UE...or_Legacy_mode
You'll want to set:
* The "OS to boot by default" to /dev/sda9
* Check box for "Separate /boot/efi partition" and set that to /dev/sda1
* Hit "Apply" when done.
Reboot and see if it worked.
If that didn't solve problem, then might be easiest to delete Linux partitions and start all over. See if this tutorial will give you a better idea on how to do the install: Guideline for Installing Mint on a Windows 8/8.1 Computer