Question

Find the missing length. If necessary round to the nearest tenth.

Short answer

Length of the missing side is $\mathbf{9}\mathbf{.}\mathbf{2}$units.

Step-by-step solution

Step 1. State the concept of ‘Pythagoras Theorem’.

Consider a right triangle with perpendicular $a$, base and hypotenuse$c$.

In right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.That is,

$c^2=a^2+b^2$

Step 2. State the concept of ‘rounding to nearest tenth’.

Suppose there are more than one digit after decimal then we round up to the $1^{st}$decimal number which is called as the tenths digit using the following rules.

Note: The number after the tenths digit is called as hundredths digit.

If the hundredths digit is less than 5, then keep the tenths digit unchanged and rewrite the number by removing decimal digits after tenths.

If the hundredths digit is greater than or equal to 5, then add 1 to the tenths digit and rewrite the number by removing decimal digits after tenths.

For example:

$6.57$is rounded to $6.6$. As the hundrendths digit is 7, which is greater than 5. The tenths digit will increase by 1.

$6.53$is rounded to $6.5$. The tenths digit 5 is kept unchanged as the hundredths digit 3 is less than 5.

Step 3. Calculate the missing length.

Observe the figure given below.

In the figure as one of the angle is 90 degree,the given triangle is a right angle triangle.

From the figure, the length of hypotenuse is 10 units and the length of perpendicular is 4 units and the length of the base is $‘b’$.

Substitute 4 as a and 10 as $c$in $c^2=a^2+b^2$, to find the missing length.

$\begin{array}{l}{c}^2=a^2+b^2\\ {\left(10\right)}^2={\left(4\right)}^2+{\left(b\right)}^2\\ 100=16+b^2\\ 100-16=b^2\\ 84=b^2\\ b^2=84\\ \text{Take\hspace{0.17em}\hspace{0.17em}square\hspace{0.17em}\hspace{0.17em}root}\\ b=\pm 9.16\\ b=\pm 9.2\left[\text{Rounding\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}nearest\hspace{0.17em}\hspace{0.17em}tenth}\right]\end{array}$

Step 4. State the conclusion.

There are two values of $‘b’$. One is role="math" localid="1647925783494" $b=-9.2$and the other one is role="math" localid="1647925778633" $b=9.2$.

As length cannot be negative, $b\ne -9.2$.

Therefore, $b=9.2$.

Hence the length of the missing side rounded to nearest tenth is $9.2$units.