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[tex]Answer: \\ PS \: is \: the \: bisector \\ = > \frac{QS}{QP} = \frac{RS}{RP} \\ < = > \frac{5}{3x + 4} = \frac{4}{2x + 3} \\ < = > 10x + 15 = 12x + 16 \\ < = > x = - \frac{1}{2} \\ = > QP = - \frac{3}{2} + 4 = \frac{5}{2} = 2.5 \\ = > RP = - 1 + 3 = 2[/tex]
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Rewritten by : Barada
The length of the side QP is 5/2 and PR is 3.
Two triangles are similar if their corresponding angles are equal and their corresponding sides are within the same ratio (or proportion).
According to the property of two similar triangles, the sides are in the same ratio.
[tex]\rm \dfrac{QS}{QP}=\dfrac{RS}{RP}\\\\\dfrac{5}{3x+4}=\dfrac{4}{2x+3}\\\\5(2x+3)=4(3x+4)\\\\10x+15=12x+16\\\\12x-10x=15-16\\\\2x=-1\\\\x=-\dfrac{1}{2}[/tex]
Substitute the value of x;
[tex]\rm QP = 3x+4 = 3 \times \dfrac{-1}{2}+4=-\dfrac{3}{2}+4 = \dfrac{-3+8}{2} =\dfrac{5}{2}\\\\ PR = 2x+3 = 2 \times \dfrac{-1}{2}+4=-1+4=3[/tex]
Hence, the length of the side QP is 5/2 and PR is 3.
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