Question

I need help on this problem

Answer 1

In the given right-angled triangle with a height n, hypotenuse m, and base
`\(2√(3)\)`, where the angle between the height and hypotenuse is 60 degrees, the values of n and m are found to be n = 6 and
`\(m = 4√(3)\)`. The correct option is (e)
`\(4√(3)\).`

In the given right-angled triangle with a height n, hypotenuse m, and base
`\( 2√(3) \)`, the angle between the height and hypotenuse is given as 60 degrees. We can use trigonometric ratios to find the values of n and m.

The trigonometric ratio for the sine of an angle in a right-angled triangle is given by:

`\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \]`

In this case, for the angle of 60 degrees:

`\[ \sin(60^\circ) = (n)/(m) \]`

Since
`\(\sin(60^\circ) = (√(3))/(2)\)`, we have:

`\[ (√(3))/(2) = (n)/(m) \]`

Solving for n, we get:

`\[ n = (m√(3))/(2) \]`

Now, using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides:

`\[ m^2 = n^2 + (2√(3))^2 \]`

Substitute the expression for n:

`\[ m^2 = \left((m√(3))/(2)\right)^2 + (2√(3))^2 \]`

Solve for m:

`\[ m^2 = (3m^2)/(4) + 12 \]`

Combine like terms:

`\[ (m^2)/(4) = 12 \]`

Multiply both sides by 4:

`\[ m^2 = 48 \]`

`\[ m = √(48) = 4√(3) \]`

Now that we have m, we can find n:

`\[ n = (m√(3))/(2) = (4√(3) * √(3))/(2) = 2 * 3 = 6 \]`

So, the correct values are:

n = 6

`\[ m = 4√(3) \]`

Now, let's find the base:

`\[ \text{Base} = 2√(3) \]`

Now, check the given options:

a. 1 (Not correct)

b. 2 (Not correct)

c. 4 (Not correct)

d.
`\(2√(6)\)` (Not correct)

e.
`\(4√(3)\)` (Correct)

Therefore, the correct answer is (e)
`\(4√(3)\)`.