The height of a triangle is 4 cm more than its base. The area is 30… Step by Step

Step by step

The question

Question: The height of a triangle is 4 cm more than its base.

The area is 30 square cm.

Find the base and the height.

1 Find what we know

What do we know?

The height is 4 more than the base.

The area is 30.

Area of a triangle = half of base times height.

What do we want?

The base and the height.

2 Name the unknown

We know the height from the base.

So we call the base x.

x = base

The height is x + 4.

x + 4 = height

Let's draw it.

The base is x.

The height is x + 4.

3 Write the equation

In words: half of base times height is the area.

12x(x + 4) = 30

So the equation is one half times x times (x + 4) = 30.

There is x times x in it: a quadratic equation.

Now we solve it.

12x(x + 4) = 30

First we remove the half: times 2 on both sides.

Half of something, times 2, is the whole thing.

30 times 2 is 60.

x(x + 4) = 60

We open the bracket: x² + 4x = 60.

x² + 4x = 60

4 Get 0 on one side

To solve it, we want 0 on the right side.

So we take away 60 on both sides.

We do it on both sides, so they stay equal.

x² + 4x − 60 = 60 − 60

Now we have x² + 4x − 60 = 0.

x² + 4x − 60 = 0

5 Factor

We need two numbers.

They multiply to −60, and they add to 4.

? × ? = −60

? + ? = 4

They multiply to a minus number, so one number is plus and one is minus.

They add to a plus number, so the bigger one is plus.

× is − → one +, one −; bigger one +

We list the pairs that make 60, smallest first.

We take the smaller from the bigger.

We need 4.

6 and 10 give 4.

Yes!

With the signs: −6 and 10.

1 × 60: 60 − 1 = 59

2 × 30: 30 − 2 = 28

3 × 20: 20 − 3 = 17

4 × 15: 15 − 4 = 11

5 × 12: 12 − 5 = 7

6 × 10: 10 − 6 = 4 ✓

So it is (x − 6)(x + 10) = 0.

(x − 6)(x + 10) = 0

6 One bracket is 0

Two numbers multiply to 0.

That only happens if one of them is 0.

So one bracket must be 0.

A × B = 0 → A = 0 or B = 0

We make each bracket 0.

Two small equations.

x − 6 = 0 or x + 10 = 0

In x − 6 = 0, we add 6 on both sides.

In x + 10 = 0, we take away 10 on both sides.

x − 6 + 6 = 0 + 6 or x + 10 − 10 = 0 − 10

So x = 6 or x = −10.

x = 6 or x = −10

7 Answer the question

x = −10: a length cannot be negative.

So we throw it away.

x = −10 ✗

a length cannot be negative

Careful!

Both numbers make the equation true.

But only one makes sense in the story.

Always look back at the story.

Careful: x = −10 fits the equation, not the story

x = 6 works: a base.

x = 6 ✓

a base

The height is x + 4: 6 plus 4 is 10.

height = 6 + 4 = 10

The base is 6 cm and the height is 10 cm.

8 Check

Let's check in the story.

Half of 6 times 10 is 30, the area.

½ × 6 × 10 = 30

And 10 is 6 plus 4, like the story says.

So the answer is right!

10 = 6 + 4

Let's look back.

First: find what we know.

Then: name the unknown.

Then: write the equation.

Then: solve it.

Then: answer the question.

Last: check.

Answer