# A wire is attached to the top of a flag pole. The wire meets the ground 4 meters from the flag pole. The wire makes a 45° angle with the flag pole. How tall is the flag pole?

Jan 30, 2018

The high of flag pole is $4 m e t e r s$

#### Explanation:

We know that tan 45º = 1

In the right triangle We need to find h and b is 4 m.
The oposite angle to h is 45º. Thus
$\tan 45 = \frac{h}{4} = 1$ by definition of $\tan \theta = \frac{h}{b}$
Trasposing we find $h = 4 m$

Jan 30, 2018

The flagpole is $4$ meters tall.

#### Explanation:

The wire, the flagpole, and the ground make up the three sides of a triangle.

• One angle of this triangle is given as ${45}^{o}$.
• The angle of the flagpole with the ground must be ${90}^{o}$.
• So the last angle must be ${45}^{o}$.

Because of the right angle $\left({90}^{o}\right)$ of the flagpole and the ground, this is a $\text{right triangle}$. The wire forms the hypotenuse.

Because both of the other angles are the same (both are ${45}^{o}$), this is an $\text{isosceles}$ right triangle (the legs are equal.)

This particular kind of right triangle is called a $45 - 45 - 90$ triangle, named for the degrees in its angles.

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The measurements of these triangles are known by memorization:

The $45 - 45 - 90$ isosceles right triangle has these properties:

• It forms half a square, cut in half along the diagonal, so it is a
$\text{right triangle}$.
• This diagonal line forms the $\text{hypotenuse}$ of the triangle.
• The sides $s$ (the legs) are the same as each other, so
it is an $\text{isosceles right triangle.}$
• The length of the hypotenuse is always $s \sqrt{2}$

Here is a diagram of the $45 - 45 - 90$ isosceles right triangle:

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Note:
Taking five minutes now to memorize this information will save you hours.

On timed standardized tests such as the SAT, ACT, or GRE, you can complete many problems in seconds while other students are still burning up their minutes solving the Pythagorean Theorem.

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In this problem, one of the sides (the side along the ground) is Given as $4$ meters.

Because both of the sides are the same, then the other side (the flagpole) must also be $4$ meters.

The flagpole is $4$ meters tall.

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Extra credit:
The wire is the hypotenuse, whose length is $s \sqrt{2}$.
So the hypotenuse is $4 \sqrt{2}$ meters long

Feb 1, 2018

The flag pole is $4$ metres high.

#### Explanation:

We need to make an assumption here - that the flagpole is perpendicular to the horizontal ground.

If one angle is 90°and the other is 45°, then the remaining angle is 45°.

45°+45°+90° = 180°

Two of the angles are equal so this is an Isosceles triangle, in fact, a right-angled isosceles triangle.

In an isosceles triangle, there are two equal sides and two equal angles opposite the equal sides.

There are no calculations necessary at all - you just need to recognise what type of triangle it is.

The equal sides are the arms of the 90° angle.
They are both $4$ metres long.

The wire from the top of the pole to the ground represents the hypotenuse. It will be longer than the other sides.