Question

To check if a doorframe is truly square (has right angles in the corners), a carpenter made the measurements shown on the image.Based on these measurements, would you agree of disagree with the following statement, explain your reasoning or show your work?(Remember to allow for measurement and rounding errors.)The top right angle of the door frame is a right angle.

Answer 1

From the doorframe we can take a triangle with the corresponding measures

we realize, it is like a right triangle then if we use pythagoras and the measures match the red angles and the other angles of the doorframe must be right

we use the hight(81) and width(32.5) to find the hypotenuse (87)

Pythagoras

`a^2+b^2=h^2`

where a and b are legs of the triangle and h the hypotenuse

then repalcing

`81^2+32.5^2=h^2`

simplify

`\begin{gathered} 6561+1056.25=h^2 \\ h^2=7617.25 \end{gathered}`

and solve for the hypotenuse

`h=\sqrt[]{7617.25}\approx87.3`

the measure of the hypotenuse is 87.3in and the measure of the diagonal of the door is 87in.

By allow of measurement and rounding errors we can say the values are the same, then the corners of the door are right and the doorframe is square