Question

NO LINKS!!!! URGENT HELP PLEASE!!!

Please help find the value of each variable for #8 for Special Triangle: 30-60-90

Answer 1

x = 23.09 (2 d.p.)

y = 11.55 (2 d.p.)

Step-by-step explanation:

The interior angles of a triangle sum to 180°. Therefore, from inspection of the given diagram, the interior angles of the given right triangle are:

• 30°, 60° and 90°.

This means the triangle is a special right triangle and that the 30-60-90 theorem can be applied to quickly find the measures of the missing sides.

What is a 30-60-90 triangle?

A 30-60-90 triangle is a special right triangle where the measures of its sides are in the ratio 1 : √3 : 2.

Therefore, the formula for the ratio of the sides is b : b√3 : 2b where:

• b = the shortest side opposite the 30° angle.
• b√3 = the side opposite the 60° angle.
• 2b = the longest side (hypotenuse) opposite the right angle.

From inspection of the given triangle, the side opposite the 60° angle is 20. Therefore:

`\implies b√(3)=20`

`\implies b=(20)/(√(3))`

`\implies b=(20√(3))/(3)`

Substitute the found value of b into the expressions for the other two sides to find the values of x and y.

x is the side opposite the right angle. Therefore:

`\implies x=2b`

`\implies x=2\cdot (20√(3))/(3)`

`\implies x=(40√(3))/(3)`

`\implies x=23.09\; \sf units\;(2\;d.p.)`

y is the side opposite the 30° angle. Therefore:

`\implies y=b`

`\implies y=(20√(3))/(3)`

`\implies y=11.55\; \sf units\;(2\;d.p.)`

Solution

• x = 23.09 (2 d.p.)
• y = 11.55 (2 d.p.)

Answer 2

`x = (40√(3))/(3)\\\\y = (20√(3))/(3)`

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Step-by-step explanation:

The short leg is y because it is opposite the 30 degree angle.

The connection between the long leg and short leg is through this formula

`\text{long leg} = (\text{short leg})*√(3)`

which applies to 30-60-90 triangles only.

Let's use that template to solve for y.

`\text{long leg} = (\text{short leg})*√(3)\\\\20 = y√(3)\\\\y = (20)/(√(3))\\\\y = (20√(3))/(√(3)√(3))\\\\y = (20√(3))/(√(3*3))\\\\y = (20√(3))/(√(9))\\\\y = (20√(3))/(3)\\\\`

Then to find the hypotenuse x, we double the short leg.

`x = 2y\\\\x = 2*(20√(3))/(3)\\\\x = (2*20√(3))/(3)\\\\x = (40√(3))/(3)\\\\`

This formula works for 30-60-90 triangles only.

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Side notes:

`(20√(3))/(3) \approx 11.54701\\\\(40√(3))/(3) \approx 23.09401\\\\`