NotThatSmart

Well, it's been a while since I've answered a question on this website, and what better way to celebrate with a geometry question :)

So, we don't have a right triangle, so we're gonna have to settle with Law of Cosines. 

Also, note that we have 

$(cos PMQ) = (- cos PMR )$

Now, let's apply Law of Cosines and add subsitute using equation above. 

$PQ^2 = QM^2 + PM^2 - 2(QM * PM) * (-cos PMR) \\ PR^2 = RM^2 + PM^2 - 2 (RM * PM) * ( cos PMR)$

Simplify

$5^2 = 5.5^2 + PM^2 + 2(5.5 * PM)cos(PMR) \\ 8^2 = 5.5^2 + PM^2 - 2(5,5 * PM) cos (PMR)$

Now, we're stuck...but we can add these two equations to keep progressing. 

$5^2 + 8^2 = 2 * 5.5^2 + 2PM^2 \\ 89 = 60.5 + 2PM^2 \\ 28.5 / 2 = PM^2\\ 14.25 = PM^2 \\ \sqrt {14.25} = PM ≈ 3.77$

And we're done!

Also, side note, this is gonna be my 1000 answer on this website which is a weird flex but whatever. 

I might be gone for another while, but hopefully be back in the summer. 

~NTS

This problem actually has a very simple answer. 

Let's note the two sidelengths given, We have $WX = 2$  and $XY = 3$

In order to find the area of WXYZ, we must find the height and base. 

The base of the rectange is $WX$ and the height of rectangle is $XY$

The problem gave all the dimensions we need to already solve this problem. We have

$WX \cdot XY = 2 \cdot 3 = 6$

So our final answer is 6. 

Thanks! :)

There are a few ways you can tackle this problem. 

The first way is reccomended. We find that after distributing, how many pairs create a coefficient of x. 

We first have x and 14. $14 \cdot x = 14x$

Next, we have 1 and -23x. $1 \cdot -23 = -23x$

These are the only two ways to achieve x. Now, we simply add then up. 

$14x - 23x = -9x$

Orrrrrrrrrrrrrrrrrrrrrrrrrrr, you can distribibute EVERYTHIG and take 100 years, but here u go :)

$x^{7}-7x^{6}+10x^{5}-13x^{4}+8x^{2}-9x+14$

Thanks! :)

According to the Triangle Angle Sum Theorem, the sum of all interior angles of a triangle is always 180 degrees. 

Using this information, we can write the equation

$2x + 3y + 8x + 15y + 4x - 2y = 180\\ 14x + 16y = 180\\ 7x+8y = 90$

Now, if we were to say that x and y can be any number, then there an infinite amount of choices. 

However, if we were to only limit x and y to integers, then $x=6; y= 6$ is the only solution. 

Hope this helps. 

~NTS

First, let's calculate the distance Will travels before grace even leaves. 

Since he traveled for 45 minutes, we have

$(45 min)(50 m/min) = 2250 m$

So when Grace starts, she and Will have

$(2800 m) – (2250 m) = 550 m$ to cover. 

Let's let t be the amount of time it takes for the two to catch up to each other after Grace starts rowing. 

We have the equation

$(50)(t) + (30)(t) = 550 \\ 80t = 550 \\ t = 550/80 = 6.875$

This is approximately 6 minutes and 52 seconds. 

So the clock time they meet is 6 min 52 sec after Grace starts.

Grace started at 2:45, so add 6 min 52 sec and the clock will read 2:51:52 pm when they meet  

Thus, our final answer is $2:51:52$

Thanks! :)

So for this problem, let's use the arithmetic series formula.

We find the sum is $\frac{(140+1)(70)}{2}$, which is $141*35.$ Now that we have a factorization, we can easily find the greatest prime divisor.

The prime factorization is $3*5*7*47$, so the greatest prime divisor is 47.

Thanks! :)

Let's know that XZW will form a triangle

$\angle ZXW = 90^\circ \\ \angle XWZ = 60^\circ$

Thus, $\angle XZW = 180 - 90 - 60 = 30$

30 degrees is our answer. 

Thanks! :)

An obtuse angle is an angle that measures in between $90^\circ$ and $180^\circ$

Here's an image that lists all types of angles.

Notice that B is the only one that fits for the obtuse angle. 

A would be an acute angle since it is less than 90 degrees. 

C and D would be reflexive angles since they are more than 180 degrees, less than 360 degrees. 

The hour hand will move at a rate of 

$360°  \space \text{in} \space 12 \space \text{hours} \\ 30° \space \text{in}  \space 1 \space \text{hour} \\ 30°/ 60  =  0.5° \space \text{in}  \text{ one minute}$

So  at   4:15, the hour  hand has  moved   $4 (30°) +  15(.5°)  =  [120+ 7.5]° = 127.5°$from 12 o'clock

The minute hand  will move  at a rate of $360°/ 60  = 6°$ per minute

So at 4:15 it will have moved  $[15* 6]°  = 90°$  from the top of the hour

So...the smaller angle formed by the hands$= 127.5 - 90 = 37.5^\circ$

May have made an error....

Thanks! :)