You are in the center of a square room and 25 ft away from a corner. What’s the area of the room?

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[johnnyragadoo]

Easier mental math - from the center to the corner is 25 and also to all corners. So, From the center to two corners forms an isosceles right triangle which is 1/4 of the square. Base times height is 25*25 or 625. The square is twice that value (or (625/2) * 4), or 1250.

[Kualinar]

You are at the centre of a square, 25' from a corner, so, 25' from each corners. Take any two adjacent corners, and you have a right triangle, with one face of the square as it hypotenuse. So, take 25^2 * 2 to get the square of length of the hypotenuse. So, 25^2 * 2 = 625 * 2 = 1250. You don't need to find the square root of 1250 just to square it. Meaning that you already have the area of the square room : 1250 square feet.

[DrDoom]

I've always liked Trig and I went the Trig route:
Sin (45) = .7071
Sin x = opp/hyp
.7071 = opp/25, opp=25(.7071)
opp = 17.67*2 (for the full side of the square)
35.35 * 35.35 (area of a square)
= 1249.811
a little more precise than just 1250, which if you're cutting wood, the .19 is about the width of a saw blade(thin kerf) and can make a difference

[erembald]

The diagonal of the square room is 50ft corner to corner. Therefore 2 x side^2 = 50^2. side^2 = area = 50^2/2 = 1250 sqft

[AnthonyWhitehorne-yl]

We are so impatient that we tend to forget that we are not the only one in the room. I already know most of what is discussed on this Channel, but I realize that he is catering for the whole class. It's the mark of a good teacher. He makes it easier for the slower one. Let's therefore remember those whose struggle with these concepts.

[martinwalker]

Triangle, area of = base times height divided by 2.
You are 25 feet from the 3 corners of a right triangle therefore the base is 50 feet and the height is 25. You have 2 of those triangles therefore the 2’s cancel, so 25 times 50 is 1250.

A second way is to recognize that a square is comprised of four right triangles that form two smaller squares that have a side length of the distance from the center of the larger square to one of its corners.
25*25= 625 is the area of each small square times 2 = 1250 sq ft.

A) youngster, I enlisted in 1971 and retired in 1995 from the Navy.
B) I fold the flag when we bury one of Uncle Sam’s Misguided Children
(For those of you that are confused, there is branch rivalry, however we band together when an outsider gets involved.)

[sharonbapp]

Like a lot of people out there I’m old and retired. Been too many years to remember this stuff. I ‘m happy for a chance to study these again. Love your explanations. Hopefully it will help keep my mind sharp

[josephatthecoop]

Draw the diagonal of the large square. The diagonal will be twice the distance from you to the corner, so 50 feet. This divides the square into 2 triangles. Choose either triangle and construct a square on each side of it. The square on the hypotenuse is 50 x 50 = 2500 square feet. By the Pythagorean theorem we know that the sum of the other two squares is also 2500. Further, the sides of the smaller squares will be equal, so their areas are also equal, and will be 2500 / 2 = 1250 square feet. Now notice that the original square had side length exactly the same as the sides of the two smaller squares. Therefore the area of the original square is exactly the same at 1250 square feet.

[borjewahlen]

Do not complain if you don't like this teacher. He is helping people that are learning math, not people that already can math.

Keep on teach, teacher 👍

[PaulTaylor-Young]

OMG! Between the poster's long-winded, repetitious verbalizing and the commenters, I have to say that I now understand why America is 37th in the world in education. I was a not a good math student in school (now 75!) but I had this figured out almost in my head in a few minutes. Too much talk and too much complicated explanations that didn't explain the basic "need to know", i.e. a sq + b sq=c sq, and c=25, sq = 625. a and b are equal, since the room is square, and a sq is half, or 312.5, and b sq is half, the sq root of which is 17.6776, so the area of one of 8 triangles (base times height) is 17.6776 squared=312.5/2 (or a + b)=156.250 x 8 (triangles) =1, 250. Some of the explanations in the comments make no sense, and by the time the poster was done yapping about incidentals, I had already stopped listening. That is the problem in school! It was the problem 60 years ago when I was in jr. high. God help our children!

[malcolmlindley]

What a long winded way of doing it! Just form the right angle in the centre, by going to an adjacent corner, not the opposite one, so you have a triangle one quarter of the square’s area. The area of this is simple: area of a triangle is half base x height, in this case 12.5x25. But as there are 4 such triangles, the area of the square is therefore 12.5x25x4, or 12.5x100, ie 1250. No need for Pythagoras’ theory etc

[robertmarino]

If a person doesn’t know the method of figuring out the area of a triangle, there’s perhaps an easier way to solve this: if you start at a corner and walk to the middle of the room, you’ve gone 25 feet; if, instead of continuing on to the _far_ corner, you turn right and go to the _next_ corner, you’ve gone another 25 feet; if *now* you imagine that you’ve just walked down two adjacent sides of an imaginary square that’s half in the room and half outside the room, then that square is 25 by 25, only half of which is in the room; 25 by 25 equals 625; you can divide that in half to get the area of the imaginary square that’s in the room, *but*, since the area of the imaginary square that’s _in_ the room takes up only one quarter of the room, we can imagine a second imaginary square of the same size that takes up another quarter of the room — in other words, don’t divide the 625 in two — — instead, we see that 625 square feet is the area of exactly *_half_* the room; so, simply double 625 to 1, 250 and you now have the total area of the room.

[ganeshdas]

Length of a diagonal of a square= a✓2 (where a is the side of the square)
As such, a✓2 = 50
a = 50/✓2 ft
Area of the square= a^2
= (50 ft/✓2)^2
1250^ft^2/2= 1250 ft^2

[richardknouse]

At age 76, I worked this out in my head. The 4 isosceles triangles form two squares with side 25. So 2 x 25 x 25 = 50 x 25 = 100 x (25/2) = 100 x 12.5 = 1250

[thomasharding]

@16:40 Keep in mind that X is only approximately equal to 35.35 ft but it is exactly equal to the square root of one half the square of the hypotenuse. The area of a SQUARE is equal to one half of the square of the hypotenuse and will equal the square of one side.

[Sailoralso]

Construct your mental box at 45 degrees to your illustration In short 50X50 simple 2500 square feet, half of that is 1250. Yes I did Pythagorean Theorem as well to confirm, , and I did the hypotenuse of a 45 degree triangle is A X 1.414. 50' divided by 1.414 is about 35 (plus a little) and 35X35 (plus a little) is about 1250 Three methods. The 50X50 divided by 2 is by far the fastest easiest.

[Eewec]

Or, the really easy way to solve it; 2 x 25 x 25 as you're in the centre of a square and 25 from the corner, that means that you can divide the room into 4 diagonally and make 2 squares of side length 25. Thus if each square has a side of 25 then twice the square of 25 is your answer. Which is 1250 ft squared.

[chamberizer]

I looked at it like four triangles.
If you put two of the triangles together you have a square with sides 25 x 25.
25 x 25 = 625 Sq ft
625 x 2 = 1250 Sq ft

[LJLMETAL]

d^2/2= A. D= diagonal. A= area. The diagonal is twice as long as 25 feet. That is 50 feet. 50^2= 2, 500 feet. 2, 500 ÷ 2= 1, 250.

[topkatz]

The square room can be thought of as 4 right triangles each with a base = height = 25.
Area of each triangle is 1/2 × 25 × 25
Since there are 4 such triangles. Total Area of Room is
4 × 1/2 × 25 × 25 =
2 × 625 = 1250 FT²

So for any Square Room with the distance given from corner to center or vice-versa the area of the square is that distance given ^2 (squared) and then doubled.
So if you are in the center of a square room 10 FT from the corner. The area of the room is 2 × 10^2 = 2 × 100 = 200 FT²