Question 51105: Hello!
Here's my question:
In triangle KLM, P is the midpoint of the line segment LM.
Prove that if PL = PK = PM, angle LKM = 90 degrees.
I have been struggling on this question for quite some time now. All I found out from the web is to introduce a point O so that KPO is a line segment with P at the midpoint, and then show that KLOM is a rectangle.
Please Help! Thank You!
Found 2 solutions by Earlsdon, Thanenjayan:
Answer by Earlsdon(6294)   (Show Source):
You can put this solution on YOUR website! I'm not certain that the following constitutes a "proof" in the formal sense of the word, but it certainly provides a demonstration.
If you accept the principle that "An angle inscribed in a semi-circle is a right angle", then read the following:
The line segment LM is the diameter of a circle.
The mid-point of LM is therefore the center of this circle and PL and PM are both radii of this circle and they, of course, are equal.
Point K is a point on the circumference of the circle, not coincident with either L or M.
The segment PK is also a radius of the circle, so that:
PL = PK = PM
Segments KL and KM form the other two sides of a triangle.
The angle LKM is inscribed in a semi-circle and is, therefore, a right angle. A right angle = 90 degrees.
Answer by Thanenjayan(7) (Show Source):
You can put this solution on YOUR website! a good question!
you have the triangle KLM,where p is the midpoint of segment LM.
and PL=PK=PM,
So you can two isocele triangles named PKL and PKM where Pk=PL=PM
in the isocele triangle PKL
angle PKL = angle PLK ( PK = PL)
in the isocele triangle PKM
angle PKM = angle PMK ( PK =PM)
so in the triangle KLM
angles MLK+LKM+KML = 180 (addition of interior angles of triangle)
angles PLK+PKL+PKM+PMK = 180 ( same angles& LKM= PKL+PKM)
so
angles 2PKL+2PKM = 180 (angle PKL = angle PLK and angle PKM = angle PMK, proved)
2(PKL+PKM)= 180
PKL+PKM = 90
so LKM = 90 (proved)
Q.E.D
Note- for geometric proves, you must draw the figures, so that U easily understand the prove
U 2nd quation is can not be varified, as we don't know the length of line segment po.
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