Question

Question 1
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 measure the length PQ.
Thinking Process
(i) Firstly we find the ratio of AB in which P divides, it with the help of the relation $AP=\frac{3}{4}AB$
(ii) Secondly, we find the ratio of AC in which Q divides. It with the help of the relation $AQ=\frac{1}{4}AC$.
(iii) Now, construct the line segment AB and AC in which P and Q respectively divide it in the ratio from step (i) and (ii) respectively.
(iv) Finally get the point P and Q. After that join PQ and get the required measurement of PQ.

Answer 1

(i) Given that AB=5cm and $AP=\frac{3}{4}AB$
$AP=\frac{3}{4}AB=\frac{3}{4}\times 5=\frac{15}{4}cm$
Then, $PB=AB-AP=5-\frac{15}{4}=\frac{20-5}{4}=\frac{5}{4}cm$ [ P is any point on the AB]
$\therefore AP:PB=\frac{15}{4}:\frac{5}{4}\Rightarrow AP:PB=3:1$

(ii) Given that AC = 7 cm and $AQ=\frac{1}{4}AC$
$AQ=\frac{1}{4}AB=\frac{1}{4}\times 7=\frac{7}{4}cm$
Then, $QC=AC-AQ=7-\frac{7}{4}=\frac{28-7}{4}=\frac{21}{4}cm$ [ Q is any point on the AC]
$\therefore AQ:QC=\frac{7}{4}:\frac{21}{4}\Rightarrow AP:PB=1:3$

(iii) Steps of constructions:
1.Draw a line segment AB of length 5 cm
2.Draw a line segment AC of length 7 cm, making an angle ${60}^{\circ }$ with AB
3.Now draw angular bisector of angle BAC (ray AX) such that it makes equal acute angle with AB and AC (i.e ${30}^{\circ }$ )
4.Along AX mark 3 + 1 = 4 Points
$A_1,A_2,A_3,A_4$ such that
$A{A}_1=A{A}_2=A_2{A}_3=A_3{A}_4$
5. Join $A_{4}B$
6. From $A_3$, draw $A_{3}P||A_{4}B$ meeting AB at P.
[By making an angle equal to $\angle B{A}_{4}A$ at $A_3$]
Then P is the point on AB which divides it in the ratio 3:1.
Thus AP : PB = 3 : 1
7. Join $A_{4}C$
8.From $A_1$, draw $A_{1}Q||A_{4}C$ meeting AC at Q.
[By making an angle equal to $\angle C{A}_{4}A$ at $A_1$]
Then Q is the point on AC which divides it in the ratio 1:3.
Thus AQ : QC = 1:3
9. Join P and Q and measure the length.