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[Haleluya]

1. Given: Quadrilateral W.I.S.H is a parallelogram. Answer the following
A. IF W.I = 3y + 3 and H.S = y + 13, how long is line H.S?
B. What is the perimeter and the area of the largest square that can be formed from rectangle W.I.S.H in 1.c.?

 

[Katz AI]

A. Since W.I.S.H is a parallelogram, opposite sides are congruent. Therefore, we have W.I = S.H and I.H = W.S.

If W.I = 3y + 3 and H.S = y + 13, then W.S = 3y + 3 and I.H = y + 13.

Since opposite sides are equal, we have:

W.S = I.H
3y + 3 = y + 13

Subtracting y from both sides, we get:

2y + 3 = 13

Subtracting 3 from both sides, we get:

2y = 10

Dividing by 2, we get:

y = 5

Therefore, H.S = y + 13 = 5 + 13 = 18.

B. Since W.I.S.H is a parallelogram, we know that the opposite sides are congruent and parallel. Therefore, the longer side is equal to the shorter side.

Let L be the length of the side of the largest square that can be formed from rectangle W.I.S.H. Then, we have:

L = min(W.I, I.H) = min(3y + 3, y + 13)

Substituting y = 5, we get:

L = min(3(5) + 3, 5 + 13) = min(18, 18) = 18.

Therefore, the perimeter of the largest square that can be formed from rectangle W.I.S.H is 4L = 4(18) = 72.

To find the area, we use the formula A = L^2, where L is the length of the side of the square. Therefore, the area of the largest square that can be formed from rectangle W.I.S.H is:

A = 18^2 = 324.

 

[Haleluya]

A. Since W.I.S.H is a parallelogram, opposite sides are congruent. Therefore, we have W.I = S.H and I.H = W.S.

If W.I = 3y + 3 and H.S = y + 13, then W.S = 3y + 3 and I.H = y + 13.

Since opposite sides are equal, we have:

W.S = I.H
3y + 3 = y + 13

Subtracting y from both sides, we get:

2y + 3 = 13

Subtracting 3 from both sides, we get:

2y = 10

Dividing by 2, we get:

y = 5

Therefore, H.S = y + 13 = 5 + 13 = 18.

B. Since W.I.S.H is a parallelogram, we know that the opposite sides are congruent and parallel. Therefore, the longer side is equal to the shorter side.

Let L be the length of the side of the largest square that can be formed from rectangle W.I.S.H. Then, we have:

L = min(W.I, I.H) = min(3y + 3, y + 13)

Substituting y = 5, we get:

L = min(3(5) + 3, 5 + 13) = min(18, 18) = 18.

Therefore, the perimeter of the largest square that can be formed from rectangle W.I.S.H is 4L = 4(18) = 72.

To find the area, we use the formula A = L^2, where L is the length of the side of the square. Therefore, the area of the largest square that can be formed from rectangle W.I.S.H is:

A = 18^2 = 324.

Another question,
Given: Quadrilateral P.O.S.T is an isosceles trapezoid with line O.S parallel to line P.T. Line E.R is its Median. answer the following:
A. If m angle P = 2x + 5 and m angle O = 3x - 10, what is m angle T?
B. E.R is 8.5 inches Long and one leg measures 9 inches. What is its perimeter if one of the bases is 3 inches more than the other?

 

[Katz AI]

Another question,
Given: Quadrilateral P.O.S.T is an isosceles trapezoid with line O.S parallel to line P.T. Line E.R is its Median. answer the following:
A. If m angle P = 2x + 5 and m angle O = 3x - 10, what is m angle T?
B. E.R is 8.5 inches Long and one leg measures 9 inches. What is its perimeter if one of the bases is 3 inches more than the other?

A. Since P.O.S.T is an isosceles trapezoid, we know that the base angles (angles P and T) are congruent, and the non-base angles (angles O and S) are also congruent.

If m∠P = 2x + 5 and m∠O = 3x - 10, then we have:

m∠P = m∠T (since they are congruent base angles)
2x + 5 = m∠T

Also, since O.S is parallel to P.T, we know that ∠O + ∠T = 180° (co-interior angles).

Substituting the values of ∠O and ∠P, we get:

(3x - 10) + (2x + 5) = 180

Simplifying the equation, we get:

5x - 5 = 180

Adding 5 to both sides, we get:

5x = 185

Dividing by 5, we get:

x = 37

Substituting x = 37, we get:

m∠O = 3x - 10 = 3(37) - 10 = 101

Therefore, m∠T = m∠P = 2x + 5 = 2(37) + 5 = 79.

B. Let B and b be the lengths of the longer and shorter bases of the trapezoid, respectively.

Since E.R is the median, we know that it divides the trapezoid into two congruent triangles. Therefore, we have:

E.R = (1/2)(B + b)

Substituting E.R = 8.5 and b = B - 3, we get:

8.5 = (1/2)(B + B - 3)

Simplifying the equation, we get:

17 = 2B - 3

Adding 3 to both sides, we get:

20 = 2B

Dividing by 2, we get:

B = 10

Therefore, b = B - 3 = 7.

The perimeter of the trapezoid is the sum of the lengths of all four sides. Therefore, we have:

Perimeter = B + b + 9 + 9 = 10 + 7 + 9 + 9 = 35.

Therefore, the perimeter of the trapezoid is 35 inches.