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mexico lost half its territory to which country? why did mexico lose so much of their land?

The correct answer is - The United States.

Mexico controlled the western, southwestern, and the westernmost part of the Midwest of what is now the United States. But the United States were growing stronger, and the Spanish support and power was declining, putting Mexico in a position to control very large territory which was not really able to properly defend. The United States had already started with expansionist politics, and once they saw that there's a good opportunity in taking over some of the Mexican territories, they did so, and managed to conquer them, incorporated them into their territory, and gained and even better strategical position.

The battle of Iwo Okinawa was a battle of the Japanese Imperial army military force against the U.S's military force.

On the U.S's side, they did it to attempt acess to mainland Japan, and on Japan's side they did it because it was mid WW2 and they couldn't let the allies stop their conquest to obtain China, the Korean peninsula, and the Philippine islands.

The battle went like this;  

The U.S invaded the island (in February 19, 1945) to try to get access to airfields of the island that was very near the coast (750 miles) of Japan to help plan an easier future mainland invasion into Japanese soil.The battle lasted for 5 weeks, making it an excruciatingly bloody battle, having around 7,200 people killed.The Japanese started running low on supplies so they had to surrender, U.S was victorious, the island fell into U.S forces. The U.S also then later captured Mount Suribachi in the island and planted a flag at the summit.

Casualties: around 7,200

NUMBER OF PEOPLE IN BATTLE:  70,000 U.S soldiers and 18,000 Japanese soldiers