The height of a triangle is 2 cm less than its base. The area is 24… Step by Step

Step by step

The question

Question: The height of a triangle is 2 cm less than its base.

The area is 24 square cm.

Find the base and the height.

1 Find what we know

What do we know?

The height is 2 less than the base.

The area is 24.

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 − 2.

x − 2 = height

Let's draw it.

The base is x.

The height is x − 2.

3 Write the equation

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

12x(x − 2) = 24

So the equation is one half times x times (x − 2) = 24.

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

Now we solve it.

12x(x − 2) = 24

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

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

24 times 2 is 48.

x(x − 2) = 48

We open the bracket: x² − 2x = 48.

x² − 2x = 48

4 Get 0 on one side

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

So we take away 48 on both sides.

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

x² − 2x − 48 = 48 − 48

Now we have x² − 2x − 48 = 0.

x² − 2x − 48 = 0

5 Factor

We need two numbers.

They multiply to −48, and they add to −2.

? × ? = −48

? + ? = −2

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

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

× is − → one +, one −; bigger one −

We list the pairs that make 48, smallest first.

We take the smaller from the bigger.

We need 2.

6 and 8 give 2.

Yes!

With the signs: 6 and −8.

1 × 48: 48 − 1 = 47

2 × 24: 24 − 2 = 22

3 × 16: 16 − 3 = 13

4 × 12: 12 − 4 = 8

6 × 8: 8 − 6 = 2 ✓

So it is (x + 6)(x − 8) = 0.

(x + 6)(x − 8) = 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 − 8 = 0

In x + 6 = 0, we take away 6 on both sides.

In x − 8 = 0, we add 8 on both sides.

x + 6 − 6 = 0 − 6 or x − 8 + 8 = 0 + 8

So x = −6 or x = 8.

x = −6 or x = 8

7 Answer the question

x = −6: a length cannot be negative.

So we throw it away.

x = −6 ✗

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 = −6 fits the equation, not the story

x = 8 works: a base.

x = 8 ✓

a base

The height is x − 2: 8 minus 2 is 6.

height = 8 − 2 = 6

The base is 8 cm and the height is 6 cm.

8 Check

Let's check in the story.

Half of 8 times 6 is 24, the area.

½ × 8 × 6 = 24

And 6 is 8 minus 2, like the story says.

So the answer is right!

6 = 8 − 2

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