For a circle with center point P cord XY is the perpendicular bisector

XP is hypotenuse, which obviously is the longest side but the longest leg is ZX (so the second longest side).

In a right triangle where the angles are 30°, 60°, and 90° the sides are always in the ratio \(1 : \sqrt{3}: 2\). Notice that the smallest side (1) is opposite the smallest angle (30°), and the longest side (2) is opposite the largest angle (90°). Since the ratio of the leg ZP to the hypotenuse XP is 1:2, then ZP (the shortest side) corresponds to 1 and thus is the opposite of the smallest angle 30°, which means that another leg ZX corresponds to \(\sqrt{3}\).

Hope it's clear.

Sorry Bunuel - still struggling. How did you get XZ = sqrt3/2? Apologies for been a pain.

It's not a problem at all.

Since \(\frac{XZ}{XP}=\frac{\sqrt{3}}{2}\) (from 30°, 60°, and 90° right triangle ratio) and \(XP=r=1\) then \(\frac{XZ}{1}=\frac{\sqrt{3}}{2}\) -->\(XZ=\frac{\sqrt{3}}{2}\).

Hope it's clear.

Hi Bunuel,

How did you assume ZPX is a 30-60-90 right triangle just from the ratio of ZP to XP (1:2). How can we assume in any triangle if the two sides are in the ratio 1:2, it will be a 30-60-90 triangle?

I thought we have to know we have to know that the triangle is 30-60-90 triangle beforehand to calculated the third side based on the ratio of two given sides.

Notice that since XY is the perpendicular to AP, then ZPX is a right triangle with right angle at Z. So, we have that side:hypotenuse=1:2, which means that we have 30-60-90 triangle, where the ratio of the sides is \(1:\sqrt{3}:2\).

Thanks for you reply Bunuel.

However, I still did not understand. Here we have angle XZP=90, XP=r and ZP=r/2.

We do not know that the other angles are 60 and 30 respectively. How can we use the ratio of two sides not three to conclude that it is a 30-60-90 triangle?

Should we know beforehand that it is a 30-60-90 triangle to use two sides to calculate the third one?

When we know two sides in a right triangle the third one is fixed.

We have side:hypotenuse=1x:2x --> third side = \(\sqrt{(2x)^2-x^2}=\sqrt{3}*x\), so the sides are in the ratio: \(1:\sqrt{3}:2\) --> 30-60-90 triangle.

Does this make sense?

Thanks Bunuel. Now it makes complete sense.

I missed the last part in calculating the third side using Pythagoras.

I wonder if this is just today...that I am looking at this perfectly clear explanation and still do not get it. I did a couple of minutes later. So first of - thanks for detailed explanations Bunuel and others. I just wanted to add that \((2x)^2-x^2 = 3x^2\) for those who look at the formula with a predetermined mind so focused on that formula and as a result forget to calculate this basic stuff, perhaps wondering where that \(\sqrt{3}*x\) came from. It is possible it is just me, but it often appears to me that it is not. This is one of those..."duuuhhh"s

Bunuel,

If 2 pi r = 2 pi r^2

then either r=0 or r=1

Since radius is always +ve, its safe to assume that r=1. Is that correct?

Yes, because we obviously have a circle.

Hey!

Could someone please check my calculations? I keep getting a wrong answer (I tried to solve it in a slightly different way but nonetheless the solution should be the same) for statement 1:

so, according to statement 1 --> 2pir=2pir² <=> r=1 ; for the following calculations, please see the attached image below.

(1) m°+n°=90° --> n°=90°-m°
(2) w°+p°=90°
(3)n°+p°=90°

--> (1) in (2): 90°-m°+p°=90° --> m°=p°, similarly: n°=w° ----> AXZ and XZB are similar triangles, hence, their side ratios will be equal.
--> (XZ/0.5)=(0.75/XZ) <=>2XZ=(3/4XZ) <=> XZ²=3/8 <=> XZ=0.5(3/2)^(1/2) --> XZ=2XZ=(3/2)^(1/2)

I tried the calculations again and again, but i keep getting the same wrong answer and not 3^(1/2). What did I do wrong? I know that the 30-60-90 approach is easier and probably quicker but I am still confused about what error I made in my calculations/ approach. If someone can help, please do so

Max

\(\frac{XZ}{AZ} = \frac{ZB}{XZ}\) --> \(AZ = 0.5\) and \(ZB = 1.5\), not 0.75.

\(\frac{XZ}{0.5} = \frac{1.5}{XZ}\) --> \(XZ^2 = \frac{3}{4}\) --> \(XZ=\frac{\sqrt{3}}{2}\).

Hope it helps.

Thanks a lot Bunuel! I totally missed that number-error!

hi bunuel ! can u please explain the 2nd condition how did we get 120 degree ?

The central angle which subtends arc XAY is angle XPY, which is 60+60=120 degrees.

Hope it's clear.

Hi Bunuel,

What formula have you applied to make statement 2 sufficient. Could you please explain. thanks!

(2) The length of Arc XAY = 2pi/3 --> \frac{2\pi}{3}=\frac{120}{360}*2\pi{r} --> r=1, the same as above. Sufficient.