Question

A rectangular piece of paper has a width that is 3 inches less than its length. It is cut in half along a diagonal to create two congruent right triangles with areas of 44 square inches. Which statements are true? Check all that apply.
A.The area of the rectangle is 88 square inches.
B.The equation x(x – 3) = 44 can be used to solve for the dimensions of the triangle.
C.The equation x2 – 3x – 88 = 0 can be used to solve for the length of the rectangle.
D.The triangle has a base of 11 inches and a height of 8 inches.
The rectangle has a width of 4 inches.

Answer 1

Let

x-------> the length of the rectangle

y------> the width of the rectangle

we know that

The area of the rectangle is equal to

$A=x*y$

The area of the two congruent right triangles  is equal to the area of the rectangle

$A=2*44=88\ in^{2}$

so

$88=x*y$ -------> equation A

$y=x-3$ -----> equation B

Substitute equation B in equation A

$x*[x-3]=88$

$x^{2} -3x-88=0$ --------> equation that can be used  to solve for the length of the rectangle

Using a graph tool-------> solve the quadratic equation

see the attached figure

The solution is

$x=11\ in$ -----> the length of the rectangle

Find the value of y

$88=11*y$

$y=8\ in$  ----> the width of the rectangle

Statements

case A) The area of the rectangle is $88$ square inches

The statement is True

See the procedure

Case B) The equation  $x*[x-3]=44$ can be used to solve for the dimensions of the triangle

The statement is False

Because, the equation $x*[x-3]=88$ can be used to solve for the dimensions of the triangle

case C) The equation $x^{2} -3x-88=0$ can be used to solve for the length of the rectangle

The statement is True

see the procedure

case D)The triangle has a base of $11$ inches and a height of $8$ inches

The statement is True

Because, the base of the triangle is equal to the length of the rectangle and the height of the triangle is equal to the width of the rectangle

case E) The rectangle has a width of $4$ inches

The statement is False

See the procedure

Answer 2

Hello,

A is TRUE (44 in²*2=88 in²)

B is FALSE x(x-3)=88 (the rectangle) or x(x-3)/2=44 (the triangle)

C is TRUE if x(x-3)=88==>x²-3x-88=0

D is TRUE :
x²-3x-88=0
Δ=9+4*88=361=19²
==>x= 11 or x=-8 (excluded for <0)

E is FALSE for 11*4≠88