The value of side [tex]q[/tex] is [tex]\boxed{q = 2\sqrt {14} }[/tex]. Option (b) is correct.
Further explanation:
The Pythagorean formula can be expressed as,
[tex]\boxed{{H^2} = {P^2} + {B^2}}.[/tex]
Here, H represents the hypotenuse, P represents the perpendicular and B represents the base.
If two triangles are similar to each other, then the ratio of the corresponding sides are equal.
Given:
The length of side QT is 10 and length of side TR is 4.
Explanation:
The [tex]\Delta{\text{ QST} \:{\text{and}\: \Delta{\text{RST}[/tex] are similar to each other. Therefore, the ratios of the corresponding sides are equal.
[tex]\begin{aligned}\frac{{{\text{SR}}}}{{{\text{SQ}}}}&= \frac{{{\text{ST}}}}{{{\text{QT}}}}\\\frac{q}{r}&= \frac{s}{{10}}\\10q&= rs\\\end{aligned}[/tex]
[tex]\begin{aligned}\frac{4}{s} &= \frac{s}{{10}}\\{s^2}&= 40\\\end{aligned}[/tex]
Apply Pythagoras theorem in triangle RST.
[tex]\begin{aligned}{4^2} + {s^2} &= {q^2}\\16 + {s^2} &= {q^2}\\{s^2} &= {q^2}- 16\\\end{aligned}[/tex]
Apply Pythagoras theorem in triangle QST.
[tex]\begin{aligned}{10^2} + {s^2} &= {r^2}\\100 + {s^2} &= {r^2}\\{s^2} &= {r^2} - 100\\\end{aligned}[/tex]
Substitute [tex]40[/tex] for [tex]{s^2}[/tex] in equation [tex]{s^2} + 16 = {q^2}.[/tex]
[tex]\begin{aligned}16 + 40 &= {q^2}\\56&= {q^2}\\\sqrt {56}&= q\\2\sqrt {14}&= q\\\end{aligned}[/tex]
The value of side [tex]q[/tex] is [tex]\boxed{q = 2\sqrt {14} }[/tex]. Option (b) is correct.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter:Triangles
Keywords: value of q, geometric mean theorem, similarity, Pythagoras theorem, ratio, corresponding sides.
