The ancient Egyptians had a neat way to construct right angles. They would make a long rope with knots to make 121212 equal sections.

When they stretched the rope into the triangle shown below, they always got a right angle. This was useful when planning the pyramids, for example.

If we apply the Pythagorean Theorem to the Egyptian knots we can see that 3^2 + 4^2 = 5^23
2
+4
2
=5
2
3, squared, plus, 4, squared, equals, 5, squared.
Which of the sets of three numbers listed would also let you form a right angle in this way?
Choose all answers that apply:
Choose all answers that apply:

(Choice A, Incorrect)
INCORRECT
\text{\{1, 1, 1\}}{1, 1, 1}start text, left brace, 1, comma, space, 1, comma, space, 1, right brace, end text

(Choice B, Checked, Correct)
CORRECT (SELECTED)
\text{\{51, 149, 140\}}{51, 149, 140}start text, left brace, 51, comma, space, 149, comma, space, 140, right brace, end text

(Choice C, Checked, Correct)
CORRECT (SELECTED)
\text{\{16, 63, 65\}}{16, 63, 65}start text, left brace, 16, comma, space, 63, comma, space, 65, right brace, end text

(Choice D, Incorrect)
INCORRECT
\text{\{6, 11, 8\}}{6, 11, 8}start text, left brace, 6, comma, space, 11, comma, space, 8, right brace, end text

Option B : will form a right angled triangle.

Option C : will form a right angled triangle.

Explanation:

Option A :

To determine whether the coordinates will form a right angled triangle, let us apply the Pythagorean theorem.

Thus, we have,

Since, both sides are not equal.

The set will not form a right angled triangle.

Hence, Option A is not the correct answer.

Option B :

Let us apply Pythagorean theorem.

Thus, we have,

Since, both sides are equal.

The set will form a right angled triangle.

Hence, Option B is the correct answer.

Option C :

Let us apply Pythagorean theorem.

Thus, we have,

Since, both sides are equal.

The set will form a right angled triangle.

Hence, Option C is the correct answer.

Option D :

Let us apply Pythagorean theorem.

Thus, we have,

Since, both sides are not equal.

The set will not form a right angled triangle.

Hence, Option D is not the correct answer.

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!