Two line segments AB and AC include an angle of 60 degree where AB = 5 cm and AC = 7 cm. Locate points P and Q on AB and AC, respectively such that AP=3by 4 and AQ=1by4 AC . Join P and Q and measure the length PQ.

Name: Two line segments AB and AC include an angle of 60 degree where AB = 5 cm and AC = 7 cm. Locate points P and Q on AB and AC, respectively such that AP=3by 4 and AQ=1by4 AC . Join P and Q and measure the length PQ.
Uploaded: Wed, 2021-01-20 17:46
Description: Two line segments AB and AC include an angle of 60 degree where AB = 5 cm and AC = 7 cm. Locate points P and Q on AB and AC, respectively such that AP=3by 4 and AQ=1by4 AC . Join P and Q and measure the length PQ.

Two line segments AB and AC include an angle of $60^{\circ}$ where AB = 5 cm and AC = 7 cm. Locate points P and Q on AB and AC, respectively such that $AP= \frac{3}{4}AB$ and $AQ= \frac{1}{4}AC$ . Join P and Q and measure the length PQ.

Answers (1)

Solution
Given :
AB = 5 cm and AC = 7 cm
$AP= \frac{3}{4}AB\: \: \cdots 1$
$AQ= \frac{1}{4}AC\: \: \cdots 2$
From equation 1
$AP= \frac{3}{4}AB$
$AP= \frac{3}{4}\times 5= \frac{15}{4}\: \: \left [ \because AB= 5cm \right ]$
P is any point on B
$\therefore PB= AB-AP= 5-\frac{15}{4}= \frac{20-15}{4}= \frac{5}{4}cm$
$\frac{AP}{AB}= \frac{15}{4}\times \frac{4}{5}= \frac{1}{3}$
$AP:AB= 1:3$
$\therefore$ scale of a line segment AB is $\frac{1}{3}$

Steps of construction
1. Draw a line segment AB = 5 cm
2.   Now draw ray AO which makes an angle i.e., $< BAO= 60^{\circ}$
3.   Which A as centre and radius equal to 7 cm draw an arc cutting line AO at C
4.   Draw ray AP with acute angle BAP
5.   Along AP make 4 points $A_{1},A_{2},A_{3},A_{4}$ with equal distance.
6.   Join $A_{4}B$
7.   From   $A_{3}$ draw $A_{3}P$ which is parallel to $A_{4}B$ which meet AB at point P.
Then P is a point which divides AB in a ratio 3 : 1
   AP : PB = 3 : 1
8.   Now draw ray AQ, with an acute angle CAQ.
9.   Along AQ mark 4 points   $B_{1},B_{2},B_{3},B_{4}$ with equal distance.
10. Join $B_{4}C$
11. From $B_{1}$ draw $B_{1}Q$ which is parallel to $B_{4}C$ which meet AC at point Q.
Then Q is a point which divides AC in a ratio 1 : 3
   AQ : QC = 1 : 3
12. Finally join PQ and its measurement is 3.25 cm.