Question

The triangle below is equilateral. Find the length of side x to the nearest tenth.

Answer 1

`x=\sqrt{(15)/(2) }`. Another form of this answer is
`x=(√(15) )/(√(2) )`(it's the same thing)

Explanation:

Because the triangle is equilateral, all other sides of the triangle are also
`√(10)`. Also, x is the altitude of this triangle as it forms a right angle with the base and is therefore perpendicular. We know that this triangle is equilateral and in any triangle that has at least two sides that are equal(in other words isosceles), the altitude is also the median and angle bisector. A median would cut the side it reaches into half. So, x would cut the base into two equal parts. Since the base is
`√(10)`, divide it by two and both halves are
`(√(10))/(2)`. What we are left with is two right triangles, both with legs
`(√(10) )/(2)` and x. Using the Pythagorean Theorem, we know that
`a^(2)+b^(2)=c^(2)`, with
`a` and
`b` being the legs and
`c` being the hypotenuse. So, we plug in the values and get
`(√(10) )/(2) ^(2)`
`+x^(2)=√(10)^(2)`. Squaring, we get
`(10)/(4)+x^(2)=10`. This becomes
`x^(2)=10-(10)/(4)` which equals
`x^(2)=(40)/(4)-(10)/(4)`. Finally,
`x^(2)=(30)/(4)`. This is
`x^(2)=(15)/(2)`. Square rooting this we get
`x=\sqrt{(15)/(2) }`. Another form of this answer is
`x=(√(15) )/(√(2) )`

Answer 2

`\sqrt{(15)/(2) }` or 2.738

Explanation:

Let’s just look at the triangle on the top with the
`√(10)` on the top and x on the bottom. (Basically the top half to the equilateral triangle)

There is a small square in the bottom right corner, which indicates that this triangle is a right triangle. This means that we can use the Pythagorean Theorem:
`a^(2) +b^(2) =c^(2)`

We know that \sqrt{10} is our hypotenuse, and therefore our c in our equation. Let’s say that x=a in our equation. Therefore we are left to find b. However, b is half the length of the side of the original equilateral triangle. An equilateral triangle means that all three sides are the same length. Therefore our side would also be \sqrt{10} units long. However we know that b is half of that value, so b=
`(1)/(2)(√(10))` or
`(√(10) )/(2)`

Plugging these values into the equation:

x^2+ (\frac{\sqrt{10} }{2})^{2}=\sqrt{10} ^{2}

`x^(2)=√(10) ^(2)-(√(10) )/(2) ^(2)`

`x^(2) =10-(10)/(4)`

`x^(2) =(15)/(2)`

`x=\sqrt{(15)/(2) }`

This approximately equals 2.738