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Nat2105 [25]
3 years ago
12

What is the value of q?

Mathematics
2 answers:
skad [1K]3 years ago
8 0

The value of side q is \boxed{q = 2\sqrt {14} }. Option (b) is correct.

Further explanation:

The Pythagorean formula can be expressed as,

\boxed{{H^2} = {P^2} + {B^2}}.

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 \Delta{\text{ QST} \:{\text{and}\: \Delta{\text{RST} are similar to each other. Therefore, the ratios of the corresponding sides are equal.

\begin{aligned}\frac{{{\text{SR}}}}{{{\text{SQ}}}}&= \frac{{{\text{ST}}}}{{{\text{QT}}}}\\\frac{q}{r}&= \frac{s}{{10}}\\10q&= rs\\\end{aligned}

\begin{aligned}\frac{4}{s} &= \frac{s}{{10}}\\{s^2}&= 40\\\end{aligned}

Apply Pythagoras theorem in triangle RST.

\begin{aligned}{4^2} + {s^2} &= {q^2}\\16 + {s^2} &= {q^2}\\{s^2} &= {q^2}- 16\\\end{aligned}

Apply Pythagoras theorem in triangle QST.

\begin{aligned}{10^2} + {s^2} &= {r^2}\\100 + {s^2} &= {r^2}\\{s^2} &= {r^2} - 100\\\end{aligned}

Substitute 40 for {s^2} in equation {s^2} + 16 = {q^2}.

\begin{aligned}16 + 40 &= {q^2}\\56&= {q^2}\\\sqrt {56}&= q\\2\sqrt {14}&= q\\\end{aligned}

The value of side q is \boxed{q = 2\sqrt {14} }. Option (b) is correct.

Learn more:

1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.

2. Learn more about equation of circle brainly.com/question/1506955.

3. Learn more about range and domain of the function brainly.com/question/3412497

Answer details:

Grade: High School

Subject: Mathematics

Chapter:Triangles

Keywords: value of q, geometric mean theorem, similarity, Pythagoras theorem, ratio, corresponding sides.

malfutka [58]3 years ago
5 0
Triangles QST and RST are similar.  Therefore, the following is true:

 q       s
--- = ----   This results in 10q=rs.
 r      10

Also, since RST is a right triangle, 4^2 + s^2 = q^2.
Since QST is also a right triangle, s^2 + 10^2 = r^2.  
            4      s
Also:  ---- = -----    which leads to s^2 = 40
            s      10

Because of this, 4^2 + s^2 = q^2 becomes 16 + 40 = 56 = q^2

Then q = sqrt(56) = sqrt(4)*sqrt(14) = 2*sqrt(14) (answer)


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