Question

suppose an architect draws a segment on a scale drawing with the end points (0,0) and (3/4,9/10). the same segment on the actual structure has the end points (0,0) and (30,36). what proportion could model this situation?

Answer 1

Answer with explanation:

Distance between two points (a,b) and (c,d) on two dimensional coordinate plane is given by

`=√((a-c)^2+(b-d)^2)`

A=(0,0)

`B=((3)/(4),(9)/(10))\\\\AB=\sqrt{((3)/(4)-0)^2+((9)/(10)-0)^2}\\\\=\sqrt{(9)/(16)+(81)/(100)}\\\\=\sqrt{(2196)/(1600)}`

⇒P=(0,0) and Q= (30, 36)

`PQ=√((30-0)^2+(36-0)^2)\\\\PQ=√(900+1296)\\\\PQ=√(2196)\\\\(PQ)/(AB)=\frac{√(2196)}{\sqrt{(2196)/(1600)}}\\\\P Q=AB *√(1600)\\\\PQ=40* AB\\\\(PQ)/(AB)=40:1`

⇒Actual Structure of segment =40 × Length of Segment on Scale

Answer 2

Let the segment be represented by AB where A(0,0) =
`A(x_(1), y_(1))` and B(3/4,9/10) =
`B(x_(2), y_(2))`.

The length of the segment drawn by architect can be calculated using distance formula:

AB =
`\sqrt{}( x_(2)- x_(1))^ {2} + (y_(2)- y_(1))^ {2}`

`AB=\sqrt{(3/4-0)^(2)+(9/10-0)^(2)`

`AB=√(9/16+81/100) \\`

`AB = (6√(61))/40`

Similarly, Let the actual end points of segment be AC where A(0,0) =
`A(x_(1), y_(1))` and C(30,36) =
`C(x_(2), y_(2))`.

The length of the original segment can be calculated using distance formula:

AC =
`\sqrt{}( x_(2)- x_(1))^ {2} + (y_(2)- y_(1))^ {2}`

`AC=\sqrt{(30-0)^(2)+(36-0)^(2)`

`AC=√(900+1296) \\`

`AC = (6√(61))`.

Thus, the actual length is 40 times the length of the segment drawn by the architect.

Thus, the proportion of the model is 1:40