Right Triangle Calculation: Finding Hypotenuse with Area 30 cm² and Base 5

Q: Why can't I use the area formula directly with the hypotenuse?

The area formula Area = ½ × base × height only works with the two perpendicular sides (legs) of a right triangle. The hypotenuse is diagonal, not perpendicular to either leg!

Q: How do I know which side is the base and which is the height?

In a right triangle, it doesn't matter! The base and height are the two perpendicular legs. You can call either one the base - just make sure you're using the two legs (not the hypotenuse) in your area calculation.

Q: What if I get a decimal when taking the square root?

Sometimes the hypotenuse will be a perfect square like $ \sqrt{169} = 13 $. If not, you can leave it as a square root or use a calculator for the decimal approximation.

Q: Can I solve this problem in a different order?

No! You must follow this sequence: 1) Use area to find the missing leg, then 2) Use Pythagorean theorem to find hypotenuse. You can't find the hypotenuse without knowing both legs first.

Q: How do I remember the Pythagorean theorem?

Remember $ a^2 + b^2 = c^2 $ where c is always the hypotenuse (longest side) and a and b are the two legs. The hypotenuse is opposite the right angle.

Right Triangle Hypotenuse with Given Area

The area of the triangle ABC is 30 cm².

What is the length of the hypotenuse?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine the length of the hypotenuse AC

00:03 Apply the formula for calculating the area of a triangle

00:06 (height(AB) x base(BC)) divided by 2

00:12 Substitute in the relevant values and proceed to solve

00:21 Multiply by the reciprocal in order to isolate AB

00:30 This is the size of AB

00:37 Now that we have 2 sides we can apply the Pythagorean theorem

00:42 We'll use it to isolate and find AC

00:48 Substitute in the relevant values and proceed to solve

00:57 Take the square root

01:02 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The area of the triangle ABC is 30 cm².

What is the length of the hypotenuse?

Step-by-step solution

To find the length of the hypotenuse of the triangle, let's proceed with the following steps:

Step 1: Use the area formula to find the base.
Step 2: Apply the Pythagorean Theorem to find the hypotenuse.

Step 1:
The formula for the area of a right-angled triangle is:

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

Given that the area is 30 cm², and one leg (the height) is 5 cm, we can find the base:

\frac{1}{2} \times \text{base} \times 5 = 30

Solve for $\text{base}$ :

\frac{1}{2} \times \text{base} \times 5 = 30

Multiply both sides by 2:

5 \times \text{base} = 60

Divide by 5:

\text{base} = 12 \, \text{cm}

Step 2:
With base and height (legs of the triangle) known, apply the Pythagorean theorem to find the hypotenuse, $c$ :

c^2 = (\text{base})^2 + (\text{height})^2 = 12^2 + 5^2

Calculate:

c^2 = 144 + 25 = 169

Take the square root to find $c$ :

c = \sqrt{169} = 13 \, \text{cm}

Therefore, the length of the hypotenuse of the triangle is 13 cm.

Final Answer

13 cm

Key Points to Remember

Essential concepts to master this topic

Area Formula: Use Area = ½ × base × height to find missing leg
Technique: Apply Pythagorean theorem: $c^2 = 12^2 + 5^2 = 169$
Check: Verify area calculation: ½ × 12 × 5 = 30 cm² ✓

Common Mistakes

Avoid these frequent errors

Using area formula incorrectly for hypotenuse
Don't try to use Area = ½ × hypotenuse × height directly = wrong result! Area formula only works with the two perpendicular legs (base and height), not the hypotenuse. Always find the missing leg first using area, then apply Pythagorean theorem for the hypotenuse.

Practice Quiz

Test your knowledge with interactive questions

Angle A is equal to 30°.
Angle B is equal to 60°.
Angle C is equal to 90°.

Can these angles form a triangle?

FAQ

Everything you need to know about this question

Why can't I use the area formula directly with the hypotenuse?

The area formula Area = ½ × base × height only works with the two perpendicular sides (legs) of a right triangle. The hypotenuse is diagonal, not perpendicular to either leg!

How do I know which side is the base and which is the height?

In a right triangle, it doesn't matter! The base and height are the two perpendicular legs. You can call either one the base - just make sure you're using the two legs (not the hypotenuse) in your area calculation.

What if I get a decimal when taking the square root?

Sometimes the hypotenuse will be a perfect square like $\sqrt{169} = 13$ . If not, you can leave it as a square root or use a calculator for the decimal approximation.

Can I solve this problem in a different order?

No! You must follow this sequence: 1) Use area to find the missing leg, then 2) Use Pythagorean theorem to find hypotenuse. You can't find the hypotenuse without knowing both legs first.

How do I remember the Pythagorean theorem?

Remember $a^2 + b^2 = c^2$ where c is always the hypotenuse (longest side) and a and b are the two legs. The hypotenuse is opposite the right angle.

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Right Triangle Calculation: Finding Hypotenuse with Area 30 cm² and Base 5

Right Triangle Hypotenuse with Given Area

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't I use the area formula directly with the hypotenuse?

How do I know which side is the base and which is the height?

What if I get a decimal when taking the square root?

Can I solve this problem in a different order?

How do I remember the Pythagorean theorem?

🌟 Unlock Your Math Potential

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