Two line segments $AB$ and $AC$ include an angle of $60 °$ 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 of $PQ$ .

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Solution

Parallel lines:
A pair of lines with the same distance apart and never intersects at any point is called a parallel line.
Step 1:
Place the ruler, and draw a line segment $AB = 5 cm$ .
Step 2:
Place the protractor at point $A$ , measure $60 °$ and sketch a ray $AX$ from the point $A$ .
Step 3:
Put the compass, and measure $7 cm$ on the ruler. Place the compass at the point $A$ and sketch an arc on the ray $AX$ at $C$ .
Step 4:
$P$ is the point on $AB$ and it is given that $AP = \frac{3}{4} AB$ . Locate the point $P$ such that the length of the line segment $AB$ is in the ratio of $3 : 1$ .
Step 5:
Draw a ray $AY$ below the line $AB$ such that it makes an acute angle $∠ BAY$ .
Step 6:
Divide the ray $AY$ into four equal parts, namely ${AA}_{1} = A_{1} A_{2} = A_{2} A_{3} = A_{3} A_{4} .$
Step 7:
Join the points $A_{4}$ and $B$ . Also join the points $A_{3}$ and $P$ . The line $A_{3} P$ parallel to $A_{4} B$ .
Step 8:
$Q$ is the point on $AC$ and it is given that $AQ = \frac{1}{4} AC$ . Locate the point $Q$ such that length of the line segment $AC$ is in the ratio of $1 : 3$ .
Step 9:
Draw a ray $AZ$ above the line $AB$ such that it makes an acute angle $∠ CAZ$ .
Step 10:
Divide the ray $AZ$ into four equal parts, namely ${AB}_{1} = B_{1} B_{2} = B_{2} B_{3} = B_{3} B_{4}$ .
Step 12:
Join the points $B_{4}$ and $C$ . Also join the points $B_{1}$ and $Q$ . The line $B_{4} C$ parallel to $B_{1} Q$ .
Step 14:
Join the points $P$ and $Q$ . scale the length of the line $PQ$ .
Hence, the length of $PQ$ is $3.25 cm$ .

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Two line segments AB and AC include an angle of 60 where AB = 5 cm and

AC = 7 cm. Locate points P and Q on AB and AC, respectively such thatAP=3 4AB And AQ=1 4AC respectively. Join P and Q and measure the length PQ.

Q. Two line segments AB and AC include an angle of

60^{0}

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.

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 $A P = \frac{3}{4} A B$ and $A Q = \frac{1}{4} A C$ .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 $A P = \frac{3}{4} A B$
(ii) Secondly, we find the ratio of AC in which Q divides. It with the help of the relation $A Q = \frac{1}{4} A C$ .
(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.

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Two line segments AB and AC include an angle of 60° where AB=5cm and AC=7cm. Locate points P and Q on AB and AC, respectively such that AP=34AB and AQ=14AC. Join P and Q and measure the length of PQ.

Two line segments $AB$ and $AC$ include an angle of $60 °$ 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 of $PQ$ .