Question

Find the length of side x to the nearest tenth.

Answer 1

Given:-

• A right angled triangle is given to us .
• Two angles are 45° , longest side is 5 and another side is "x" .

To find:-

• The value of x .

Answer:-

In the given right angled triangle, we may use the trigonometric ratios. We can see that the measure of the longest side is 5 which is hypotenuse and the measure of perpendicular needs to be find out .

We may use the ratio of sine here as , we know that in any right angled triangle,

`\implies\sin\theta =(p)/(h) \\`

And here , p = 5 and h = x , so on substituting the respective values, we have;

`\implies \sin\theta = (x)/(5) \\`

Again here angle is 45° . So , we have;

`\implies \sin45^o =(x)/(5) \\`

The measure of sin45° is 1/√2 . so on substituting this we have;

`\implies (1)/(\sqrt2)=(x)/(5) \\`

`\implies x =(5)/(\sqrt2)\\`

Value of √2 is approximately 1.414 . So we have;

`\implies x =(5)/(1.414) \\`

`\implies \underline{\underline{\red{\quad x = 3.53\quad }}}\\`

Hence the value of x is 3.53 .

and we are done!

Answer 2

The length of side x to the nearest tenth is 3.5.

Explanation:

From inspection of the given right triangle, we can see that the interior angles are 45°, 45° and 90°. Therefore, this triangle is a 45-45-90 triangle.

A 45-45-90 triangle is a special right triangle where the measures of its sides are in the ratio 1 : 1 : √2. Therefore, the formula for the ratio of the sides is b: b : b√2 where:

• b is the measure of the legs opposite the 45° angles.
• b√2 is the longest side (hypotenuse) opposite the right angle.

We have been given the hypotenuse, so:

`\implies b√(2)=5`

Solve for b:

`\implies (b√(2))/(√(2))=(5)/(√(2))`

`\implies b=(5)/(√(2))`

The side labelled "x" is one of the sides opposite the 45° angles, so:

`\implies x=b`

Substitute the found value of b into the equation for x:

`\implies x=(5)/(√(2))`

`\implies x=3.5355339...`

`\implies x=3.5\;\sf(nearest\;tenth)`

Therefore, the length of side x to the nearest tenth is 3.5.