Question

In the figure, lines l, m and n are parallel. AP = PQ. Find

1. BC if AB = 3.5 cm
2. SQ if RQ = 2.8 cm
3. CQ if BP = 3.2 cm
4. PR if AS = 7.4 cm
5. BR if AS = 6.5 cm and CQ = 9.5 cm

Answer 1

i. Find BC if AB = 3.5 cm

On left transversal `AC: (AB)/(BC)`

= `(AP)/(PQ)`

= 1

⇒ AB = BC

\[\boxed{\text{BC = 3.5 cm}}\]

ii. Find SQ if RQ = 2.8 cm

Right vertical S – R – Q is a transversal. With equal spacing, SR = RQ = 2.8.

So SQ = SR + RQ

= 2.8 + 2.8

= \[\boxed{\text{5.6 cm}}\]

iii. Find CQ if BP = 3.2 cm

Between the two slant transversals (left and right sides), the widths on m and n are in the ratio of their levels:

BP : CQ = (height `l → m`) : (height `l → n`) = 1 : 2.

So CQ = 2 × BP

= 2 × 3.2

= \[\boxed{\text{6.4 cm}}\]

iv. Find PR if AS = 7.4 cm

Consider the two transversals: right slant AQ and right vertical SQ. They meet at Q. Bases on parallels are AS (on l), PR (on m), QR (= 0 on n at the vertex). With equal spacing,

`(PR)/(AS) = ("height"  Q → m)/("height"  Q → l) = 1/2`

⇒ `PR = (AS)/2`

= `(7.4)/2`

= \[\boxed{\text{3.7 cm}}\]

v. Find BR if AS = 6.5 cm and CQ = 9.5 cm

Between transversals “left slant” and “right vertical”, the widths on l, m, n are in arithmetic progression (linear with level).

Hence the middle width equals the average:

`BR = (AS + CQ)/2`

`(6.5 + 9.5)/2`

= \[\boxed{\text{8 cm}}\]