Be patient. I have not fully understood
Why 5/2 * PQ...should be QS ??
Secondly why my reasoning do not lead me to the answer and is flawed, completely wrong ???
We have a right triangle PQS where angle P is 90. Also two sides PS=3 and PQ= 4 so from this we have a 30-60-90 triangle.
So, if we look at angle R is 90, from this we can see that angle P is shared between PQS and PRS so P for PRS should be 60 (likewise angle P for triangle PRQ should be 30, so angle P is 60+30=90).
So, we have: for triangle PRS angle R is 90 (PS is the hypotenuse), angle P is 60 (RS long leg) and angle S is 30 (short leg PR).
If PS is 3 (hypotenuse opposite 90 angle) PR should be 1.5 (short leg opposite angle S that is 30). This based on ratio 30:60:90.
I know that it does not hold anywater, but is useful to understand
Thanks
First of all 5/2 * PQ does not equal to QS. We equate the areas, which can be found in two ways:
1. 1/2*Leg1*Leg2 --> \(area=\frac{1}{2}*PQ*PS=6\);
2. 1/2*Perpendicular to hypotenuse*Hypotenuse --> \(area=\frac{1}{2}*PR*QS=\frac{5}{2}*PR\) (since hypotenuse QS=5);
Now, equate the areas: \(6=\frac{5}{2}*PR\) --> \(PR=\frac{12}{5}\).
Next, in a right triangle where the angles are 30°, 60°, and 90° the sides are always in the ratio \(1 : \sqrt{3}: 2\). In PQS sides PQ and PS are NOT in the ratio \(1 : \sqrt{3}\), so PQS is not a 30°, 60°, and 90° right triangle. I think you are mixing 3-4-5 Pythagorean Triples triangle with 30°-60°-90° triangle.