Plane Geometry Problem

In summary, the conversation discusses a problem in which a point T is taken on the line HM in an acute-angled triangle PQR with angle P=\pi/6, and it is shown that PT=2QR by connecting points T and R and points T and Q. It is also proven that the quadrilateral HRTQ is a parallelogram and that triangle PQN is isosceles with < RNQ = 60 degrees. The solution can be found at http://www.idealmath.com/triangle.pdf.
  • #1
aniketp
84
0
Hi everyone,
Can anyone solve the followong by plane Euclidean geometry?
I got it by co-ordinate geometry, but couldn't get it by plane...
>In an acute - angled triangle PQR , angle P=[tex]\pi[/tex]/6 , H is the orthocentre, and M is the midpoint of QR . On the line HM , take a point T such that HM=MT. Show that PT=2QR.
 
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  • #2
Connect points T and R.
Connect points T and Q.
Quadrilateral HRTQ is a parallelogram because HM = MT and MR = MQ.

Thus
TR is perpendicular to PR, and
TQ is perpendicular to PQ.

Let N be the midpoint of PT.
Connect points N and R.
Connect points NQ.
In right triangle PRT, NR = PT/2.
In right triangle PQT, NQ = PT/2

So NR = NQ = PT/2
Thus triangle PQN is isosceles.

Now we will show that < RNQ = 60 degrees.
(note that < NPR = < NRP and < NPQ = < NQP)
< RNQ = < RNT + < QNT
= < NPR + < NRP + < NPQ + < NQP
= 2( < NPR + < NPQ)
= 2 < RPQ
= 2*30
= 60

So triangle NPQ is equilateral
Thus PQ = NQ

As NQ = PT/2
So PQ = PT/2
PT = 2PQ

http://www.idealmath.com
 

1. What is Plane Geometry?

Plane Geometry is a branch of mathematics that deals with the properties and relationships of points, lines, angles, and shapes on a flat surface, also known as a plane.

2. What are the basic elements of Plane Geometry?

The basic elements of Plane Geometry are points, lines, angles, and shapes. Points are represented by a dot and have no size or dimension. Lines are represented by a straight path of infinite length and have no thickness. Angles are formed by two intersecting lines and measure the amount of rotation between them. Shapes are formed by connecting points and lines together.

3. What are the different types of angles in Plane Geometry?

There are three types of angles in Plane Geometry: acute, obtuse, and right. Acute angles measure less than 90 degrees, obtuse angles measure more than 90 degrees but less than 180 degrees, and right angles measure exactly 90 degrees.

4. What is the Pythagorean Theorem and how is it used in Plane Geometry?

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem is used to solve for missing side lengths in right triangles.

5. How is Plane Geometry used in real life?

Plane Geometry is used in many real-life scenarios, such as designing buildings, creating maps, and measuring land. It is also used in fields such as engineering, architecture, and computer graphics. Understanding Plane Geometry helps us visualize and understand the world around us.

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