Question

What is the value of q?

Answer 1

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.

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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.

Answer 2

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)