In the lecture slide, the professor uses the orange box to explain how the t-test is really a special case of the ANOVA. However, the t-test variance appears to be greater than the ANOVA variance by a factor of sqrt(2) and therefore F would not equal t^2. Is her slide wrong?
It is not the one-sample t-test, which you might probably considering. It was from the different formula. Go check the "two-sample t-test". (Or you can directly go check the F-test table and the T-test table. Don't forget to use the two-tailed one.)
In fact, you better to consider more about the meaning of the test, that's why we could tell the conclusion from the beginning. The formula work is just to confirm, or to show you guys, rather than to convince everyone by words.