Steply

Right triangle with its height to the hypotenuse

Euclid's rules · Geometry · 20 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Find the similar triangles
  2. Write the rule
  3. Put the numbers in
  4. Take the square root
  5. Check

The steps for Right angle at A, AD ⊥ BC, BD = 4, DC = 9, find AD. Every solution names its steps, so you always know what comes next.

Worked examples

Right angle at A, AD ⊥ BC, BD = 4, DC = 9, find AD Easy

The question

Question: triangle A B C has a right angle at A, and A D is at 90° to B C, with BD = 4, DC = 9: find AD.

1 Find the similar triangles

The height A D cuts the big triangle into two small ones.

All three triangles have a right angle, and they share their other angles.

So they are similar: the same shape.

2 Write the rule

Triangles D B A and D A C: B D over A D equals A D over D C.

Cross-multiplied, that gives the rule: AD squared is BD times DC.

AD2 = BD × DC

Careful!

It is BD times DC, not plus.

Careful: not AD² = BD + DC ✗

3 Put the numbers in

Put the numbers in.

We call AD x.

x2 = 4 × 9

4 times 9 is 36.

x2 = 36

4 Take the square root

x is the square root of 36.

A length is positive: 6.

x = √36 = 6

5 Check

Let's check with Pythagoras in the small triangle A B D: A D squared plus B D squared is 52 the same as A B squared.

So it fits!

triangle ABD: AD² + BD² = 36 + 16 = 52 = AB² = 52

Let's look back.

First: find the similar triangles.

Then: write the rule.

Then: put the numbers in.

Then: take the square root.

Last: check.

Answer
x = √36 = 6

▶ Watch it solved

Right angle at A, AD ⊥ BC, BD = 9, BC = 25, find AB Medium

The question

Question: triangle A B C has a right angle at A, and A D is at 90° to B C, with BD = 9, BC = 25: find AB.

1 Find the similar triangles

The height A D cuts the big triangle into two small ones.

All three triangles have a right angle, and they share their other angles.

So they are similar: the same shape.

2 Write the rule

Triangles D B A and A B C: B D over A B equals A B over B C.

Cross-multiplied, that gives the rule: AB squared is BD times BC.

AB2 = BD × BC

Careful!

It is BD times BC, not plus.

Careful: not AB² = BD + BC ✗

3 Put the numbers in

Put the numbers in.

We call AB x.

x2 = 9 × 25

9 times 25 is 225.

x2 = 225

4 Take the square root

x is the square root of 225.

A length is positive: 15.

x = √225 = 15

5 Check

Let's check with Pythagoras in the small triangle A B D: A D squared plus B D squared is 225 the same as A B squared.

So it fits!

triangle ABD: AD² + BD² = 144 + 81 = 225 = AB² = 225

Let's look back.

First: find the similar triangles.

Then: write the rule.

Then: put the numbers in.

Then: take the square root.

Last: check.

Answer
x = √225 = 15

▶ Watch it solved

Right angle at A, AD ⊥ BC, BD = 3, DC = 5, find AD Harder

The question

Question: triangle A B C has a right angle at A, and A D is at 90° to B C, with BD = 3, DC = 5: find AD.

1 Find the similar triangles

The height A D cuts the big triangle into two small ones.

All three triangles have a right angle, and they share their other angles.

So they are similar: the same shape.

2 Write the rule

Triangles D B A and D A C: B D over A D equals A D over D C.

Cross-multiplied, that gives the rule: AD squared is BD times DC.

AD2 = BD × DC

Careful!

It is BD times DC, not plus.

Careful: not AD² = BD + DC ✗

3 Put the numbers in

Put the numbers in.

We call AD x.

x2 = 3 × 5

3 times 5 is 15.

x2 = 15

4 Take the square root

x is the square root of 15.

A length is positive: 3.9, to the nearest tenth.

x = √15 ≈ 3.9

5 Check

Let's check with Pythagoras in the small triangle A B D: A D squared plus B D squared is 24, about the same as A B squared.

So it fits!

triangle ABD: AD² + BD² = 15 + 9 = 24 = AB² = 24

Let's look back.

First: find the similar triangles.

Then: write the rule.

Then: put the numbers in.

Then: take the square root.

Last: check.

Answer
x = √15 ≈ 3.9

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. right angle at A, AD ⊥ BC, BD = 4, DC = 9, find AD
    Answer
    x = √36 = 6
  2. right angle at A, AD ⊥ BC, BD = 2, DC = 8, find AD
    Answer
    x = √16 = 4
  3. right angle at A, AD ⊥ BC, BD = 3, DC = 12, find AD
    Answer
    x = √36 = 6
  4. right angle at A, AD ⊥ BC, BD = 4, BC = 9, find AB
    Answer
    x = √36 = 6
  5. right angle at A, AD ⊥ BC, DC = 9, BC = 25, find AC
    Answer
    x = √225 = 15
  6. right angle at A, AD ⊥ BC, BD = 1, DC = 16, find AD
    Answer
    x = √16 = 4

Medium

  1. right angle at A, AD ⊥ BC, BD = 9, BC = 25, find AB
    Answer
    x = √225 = 15
  2. right angle at A, AD ⊥ BC, AB = 6, BC = 12, find BD
    Answer
    x = 3612 = 3
  3. right angle at A, AD ⊥ BC, AC = 12, BC = 16, find DC
    Answer
    x = 14416 = 9
  4. right angle at A, AD ⊥ BC, AD = 6, BD = 4, find DC
    Answer
    x = 364 = 9
  5. right angle at A, AD ⊥ BC, AD = 12, DC = 16, find BD
    Answer
    x = 14416 = 9
  6. right angle at A, AD ⊥ BC, BD = 5 cm, DC = 20 cm, find AD
    Answer
    x = √100 = 10 cm
  7. right angle at A, AD ⊥ BC, AB = 10, BC = 25, find BD
    Answer
    x = 10025 = 4

Harder

  1. right angle at A, AD ⊥ BC, BD = 3, DC = 5, find AD
    Answer
    x = √15 ≈ 3.9
  2. right angle at A, AD ⊥ BC, BD = 2, BC = 7, find AB
    Answer
    x = √14 ≈ 3.7
  3. right angle at A, AD ⊥ BC, AC = 6, BC = 9, find DC
    Answer
    x = 369 = 4
  4. right angle at A, AD ⊥ BC, BD = 4, DC = 6, find AD
    Answer
    x = √24 ≈ 4.9
  5. right angle at A, AD ⊥ BC, AD = 8, BD = 5, find DC
    Answer
    x = 645 = 12.8
  6. right angle at A, AD ⊥ BC, DC = 3 cm, BC = 10 cm, find AC
    Answer
    x = √30 ≈ 5.5 cm
  7. right angle at A, AD ⊥ BC, AB = 9, BC = 15, find BD
    Answer
    x = 8115 = 5.4