# Prove PM is Diameter in ΔABC with ∠A = 90°

• Lukybear
In summary, the conversation is about proving that PM is a diameter of a circle, given certain information about a triangle and a circle. The solution involves showing that certain triangles are similar and proving that the line segment AK is equal to the line segment KH.
Lukybear

## Homework Statement

In ΔABC, ∠A = 90°, M is the midpoint of BC and H is the foot of the altitude from A to BC. A circle l is drawn through points A, M and C. The line passing through M perpendicular to AC meets AC at D and the circle l again at P. BP intersects AH at K.

Prove that PM is diameter.

## The Attempt at a Solution

No idea how to do at all. Any help would be appreciated, as this is part 1 of a long question.

#### Attachments

• Circle Geo SGGS.jpg
14.5 KB · Views: 463
Last edited:
okay, but what are you trying to figure out?

Opps sorry. I want to figure out that PM is the diameter

Have found solution but need further help! Here is my attempt:

Found PM is diameter of circle l, Triangle MCD similar to Triangle MPC, Triangle DMB similar to Triangle BMP, Angle DBM = Angle ABK.

Prove AK = KH, (using similar triangles or otherwise)

There seems to be a lot of extraneous information here. To prove that PM is a diameter of circle [itex]l[/tex], all you need to do is note that you are given a right triangle ([itex]\triangle ABC[/tex]) with M as the midpoint of BC. Therefore, D is the midpoint of AC (which can be shown since [itex]\triangle ABC[/tex] is similar to [itex]\triangle DMC[/tex]). AC is a chord to circle [itex]l[/tex] and since D is the midpoint of that chord, PM has to be a diameter.

## 1. How do you prove that PM is a diameter in ΔABC?

In order to prove that PM is a diameter in ΔABC, we must show that it passes through the center of the circle and bisects the chord AB.

## 2. What is the definition of a diameter?

A diameter is a line segment that connects two points on a circle and passes through the center of the circle. It is also the longest chord of a circle.

## 3. Why is ∠A = 90° important in this proof?

∠A = 90° is important because it is one of the properties of a right triangle. By showing that PM is a diameter, we can also prove that ΔABC is a right triangle and use properties of right triangles to further prove that PM is a diameter.

## 4. What are the steps to prove that PM is a diameter in ΔABC?

The steps to prove that PM is a diameter in ΔABC are as follows:

1. Show that PM passes through the center of the circle.
2. Show that PM bisects the chord AB.
3. Use the Pythagorean Theorem to show that ΔABC is a right triangle.
4. Use the properties of right triangles to show that PM is a diameter.

## 5. Can this proof be applied to any right triangle?

Yes, this proof can be applied to any right triangle inscribed in a circle. As long as we can show that the hypotenuse of the right triangle is a diameter of the circle, we can prove that the triangle is a right triangle.

Replies
16
Views
2K
Replies
4
Views
3K
Replies
11
Views
2K
Replies
1
Views
1K
Replies
1
Views
1K
Replies
3
Views
3K
Replies
13
Views
4K
Replies
6
Views
2K
Replies
1
Views
1K
Replies
18
Views
4K