Tangent from P, radius 5, OP = 13, find PT Easy

The question

Question: find PT.

PT is a tangent to the circle with center O.

The radius is 5.

OP = 13.

1 Use the tangent rule

PT touches the circle at T, and nowhere else.

It is called a tangent.

OT goes from the center to the touch point: a radius.

The radius meets the tangent at a right angle.

This is always true for a tangent.

So O, T and P make a right triangle.

The right angle is at T.

2 Name the sides

The longest side is called the hypotenuse.

We call it c.

It is always across from the right angle.

The two shorter sides make the right angle.

We call them a and b.

The rule for every right triangle: a² + b² = c².

Careful!

OP is across from the right angle.

So OP is the longest side, c.

Not PT.

3 Put the numbers in

Now put the numbers in: a = 5 and c = 13.

4 Work out the squares

5² means 5 times 5.

That is 25.

13² means 13 times 13.

That is 169.

Write the squares: 25 + b² = 169.

5 Remove the 25

We want b² by itself.

The 25 is in the way.

To remove 25, we take away 25.

We do it on both sides.

Why both sides?

The = sign means both sides are the same amount.

If we change only one side, they are not the same any more.

25 and − 25 make zero.

Cross them out.

169 minus 25 is 144.

6 Take the square root

b² is 144.

So b times b is 144.

That means b is the square root of 144.

The square root of 144 is the positive number that times itself makes 144.

12 times 12 is 144.

So b = 12.

The missing side is 12.

7 Check

Let's check: put b = 12 back.

First a² + b².

Now c².

Both are 169.

So b = 12 is right!

Let's look back.

First: use the tangent rule.

Then: name the sides.

Then: put the numbers in.

Then: work out the squares.

Then: remove the 25.

Then: take the square root.

Last: check.

▶ Watch it solved

Tangent from P, angle at O 50, angle at P x Medium

The question

Question: find x.

PT is a tangent to the circle with center O.

The angle at O is 50°.

The angle at P is x.

1 Use the tangent rule

PT touches the circle at T, and nowhere else.

It is called a tangent.

OT goes from the center to the touch point: a radius.

The radius meets the tangent at a right angle: 90°.

This is always true for a tangent.

2 Use the angle fact

O, T and P make a triangle.

Its three angles add up to 180°.

So 50 + x + 90 = 180.

Careful!

The right angle at T is one of the three angles.

It goes in the sum too.

3 Group like terms

Terms are the parts between the + and − signs.

Like terms are the same kind: same letters, same powers.

Plain numbers go together.

50 + 90 = 140.

Now we have 140 + x = 180.

4 Remove the 140

We want x by itself.

The 140 is in the way.

To remove 140, we take away 140.

We do it on both sides.

Why both sides?

The = sign means both sides are the same amount.

If we change only one side, they are not the same any more.

140 and − 140 make zero.

Cross them out.

180 − 140 = 40.

So x = 40!

5 Check

Let's check: put x = 40 back and add the three angles.

They add up to 180°.

So x = 40 is right!

Let's look back.

First: use the tangent rule.

Then: use the angle fact.

Then: group like terms.

Then: remove the 140.

Last: check.

▶ Watch it solved

Tangent from P, PT = 12, OP = 13, find the radius Harder

The question

Question: find the radius OT.

PT is a tangent to the circle with center O.

OP = 13.

PT = 12.

1 Use the tangent rule

PT touches the circle at T, and nowhere else.

It is called a tangent.

OT goes from the center to the touch point: a radius.

The radius meets the tangent at a right angle.

This is always true for a tangent.

So O, T and P make a right triangle.

The right angle is at T.

2 Name the sides

The longest side is called the hypotenuse.

We call it c.

It is always across from the right angle.

The two shorter sides make the right angle.

We call them a and b.

The rule for every right triangle: a² + b² = c².

Careful!

OP is across from the right angle.

So OP is the longest side, c.

Not OT.

3 Put the numbers in

Now put the numbers in: a = 12 and c = 13.

4 Work out the squares

12² means 12 times 12.

That is 144.

13² means 13 times 13.

That is 169.

Write the squares: 144 + b² = 169.

5 Remove the 144

We want b² by itself.

The 144 is in the way.

To remove 144, we take away 144.

We do it on both sides.

Why both sides?

The = sign means both sides are the same amount.

If we change only one side, they are not the same any more.

144 and − 144 make zero.

Cross them out.

169 minus 144 is 25.

6 Take the square root

b² is 25.

So b times b is 25.

That means b is the square root of 25.

The square root of 25 is the positive number that times itself makes 25.

5 times 5 is 25.

So b = 5.

The missing side is 5.

7 Check

Let's check: put b = 5 back.

First a² + b².

Now c².

Both are 169.

So b = 5 is right!

Let's look back.

First: use the tangent rule.

Then: name the sides.

Then: put the numbers in.

Then: work out the squares.

Then: remove the 144.

Then: take the square root.

Last: check.

▶ Watch it solved