Question

In the following figure, l || m || n and AB = BC, find:

1. PQ if QR = 4 cm
2. BO if CR = 6 cm

Answer 1

Given:

Lines l || m || n. Points A, B, C lie on one transversal top, middle, bottom respectively and P, Q, R lie on the other transversal top, middle, bottom respectively.

AB = BC.

1. QR = 4 cm. 
2. CR = 6 cm.

i. Find PQ when QR = 4 cm

Step-wise calculation:

1. AB = BC ⇒ B is the midpoint of AC.

By the midpoint/intercept theorem for three parallel lines, the corresponding segment on the other transversal is also bisected.

So, Q is the midpoint of PR.

Hence, PQ = QR.

2. Given QR = 4 cm

And PQ = QR

⇒ PQ = 4 cm

ii. Find BO when CR = 6 cm

Step-wise calculation:

1. Put coordinates or use linear interpolation along the line AR:

For the line joining A (on l) and R (on n), the point on m height halfway between l and n has x-coordinate `xO = (xA + xR)/2`.

Likewise, for the line joining A and C, the middle point B on m has `xB = (xA + xC)/2`.

This is the same midpoint/intercept idea on parallel lines.

2. BO = |xO – xB|

= `|(xA + xR)/2 - (xA + xC)/2|`

= `|(xR - xC)|/2`

3. But |xR – xC| is the horizontal distance CR on the bottom line.

So, `BO = (CR)/2`.

4. Given CR = 6 cm

⇒ `BO = 6/2` 

⇒ BO = 3 cm