1. Area of a Triangle

Which is wider?

To begin with, please look at the Figure 1.

Figure 1

There are two groups split into the Group A and the Group B. They are playing the (1 on 3's) TAG GAME using the ball each other. Novy belongs to the Group A. Then, he said like this.

Novy "Hey, you guys! That's not fair. I can't take the ball because of this wide area!"
Jason "What did you say, Novy? Isn't the Group B wide too? You are annoying in spite of being that social position. Eh?"
Coach Dragon "OK, OK. Don't argue. And take it easy please. In that case, let's compare the Group A with the Group B and check them."

Well, which of the following from (A) to (D) is correct?

Exercise 1  
[Question]
What is the solution of the area of a triangle?

(A) It's a base times a height.
(B) It's a base times a height divided by 2.
(C) First, you prepare pasteboard, then measure weight of 1 m wide of it. Secondly, you cut it as an arbitrary triangle and measure its weight. Finally, you will find the area, if you calculate that "the weight of the triangle divided by the weight of 1 m wide".
(D) It's OK, if three sides are known.

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[Answer]
(A) Since it is how to find the area of a quadrilateral, it's wrong.
(B) It is how to find the area of a triangle. Of course, the height (or altitude) is perpendicular to the base, you know.
(C) The area is more exact as it measures exactly. However, the pasteboard will probably be wasteful. But, a person who can get such an idea may have a gift for being a scholar.
(D) Although this explanation is not enough, in fact, the area can be found by this method. We learn the thing "Heron's formula" in a high school. So if you haven't learned yet, please look forward to it until then.

Advanced Topic

"Heron's formula" is a formula to find the area of a triangle, when its three sides are known. When three sides stand for a, b, and c respectively, and a half of the sum of them {i.e., ((a + b + c)) / 2} stands for "s", the square root of (s (s - a) (s - b) (s - c) ) will be the area of the triangle. In a high school, you would learn the method how to find it other than "Heron's formula".

Heron was a geometrician from Alexandria Egypt. He was a person who lived in about 100 B.C.

Now, I examine the method of (B) we learned in an elementary school. Each the base and the height are as shown in the Figure 2.

Figure 2

I have a question.

Exercise 2  
[Question]
Find the area of the groups A and B. Then evaluate whether the Group A is wider than the other one or not.

(A) The Group A: 97.5m. The Group B: 108m. Then, the Group A is less wide.
(B) The Group A: 195m. The Group B: 108m. Then, the Group A is wider.
(C) The Group A: 97.5m. The Group B: 54m. Then, the Group A is wider.
(D) All of the answers of (A), (B), and (C) are errors.

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[Answer]
Isn't correct (C)? It's an easy problem if you can substitute the base and the height correctly, I think.

Why is "A base times a height divided by 2" ?

It's the main subject from here.

Exercise 3  
[Question]
Well, why the area of a triangle is found by "A base times a height divided by 2" ?
Please consider for a while.

(A) Because the examination result which is like (C) of Exercise 1 became clear as becoming like the formula.
(B) It is natural even if we can't understand now. We should learn the exact reason in a high school.
(C) Just because a triangle is a half of a quadrilateral.
(D) All of the answers of (A), (B), and (C) are errors.

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[Answer]
(D) is correct. (B) is wrong especially. Because you can sufficiently explain that even if you are a schoolchild. (C) is somehow correct. But the explanation is vague.
Well then, please look at the animation of the Figure 3. The scene number is displayed on the upper left.

Figure 3

Have you understood the reason? Thus, a triangle can be changed its shape into a half of a rectangle, without changing its base and its height.
If you swallowed the formula of the area of a triangle when you were in an elementary school, please understand precisely at this opportunity.

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Novy "It wasn't a lie I said. Isn't this one wider?"
Snake "What did you say? Even if it is less wide, but you can't take the ball, Novy?"
Jason "That's right. Whether it is wide or not, it's not related to you."
Novy "Boohoo. Coach Dragoooooooooon!"