Which of the following triangles have the same areas?101010121212555131313555888121212666666FFFEEEGGGCCCBBBAAAKKKJJJIIIDDDLLLHHH

All the triangles have equal areas.

Triangle Area Comparison: Analyzing Three 5-12-13 Configurations

Q: How do I know which sides to use for base and height?

In a right triangle, use the two perpendicular sides (the legs) that meet at the 90° angle. Never use the hypotenuse, which is the longest side opposite the right angle.

Q: How do I identify the right angle in each triangle?

Look for the square symbol in the corner of each triangle in the diagram. This marks the 90° angle where the two perpendicular sides (legs) meet.

Triangle Area Calculations with Right Triangles

Which of the following triangles have the same areas?

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine which of the following triangles have the same area

00:03 Apply the formula for calculating the triangle area

00:07 (height x by base) divided by 2

00:14 Substitute in the relevant values and solve to find the area

00:20 This is the area of triangle IJK

00:26 Apply the same method and calculate the areas of the other triangles

00:34 Substitute in the height and base of the triangle EFG

00:53 This is the area of the triangle EFG

01:05 Substitute in the height and base of the triangle ABC

01:16 Substitute in the height and base of triangle ABC

01:25 This is the area of triangle ABC

01:32 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Which of the following triangles have the same areas?

Step-by-step solution

We calculate the area of triangle ABC:

$\frac{12\times5}{2}=\frac{60}{2}=30$

We calculate the area of triangle EFG:

$\frac{6\times10}{2}=\frac{60}{2}=30$

We calculate the area of triangle JIK:

$\frac{6\times5}{2}=\frac{30}{2}=15$

Therefore, the triangles that have the same areas are ABC and EFG.

Final Answer

EFG and ABC

Key Points to Remember

Essential concepts to master this topic

Area Formula: Use A = (1/2) × base × height for right triangles
Technique: Identify perpendicular sides: ABC uses base 12, height 5 = 30
Check: Calculate each triangle area and compare results directly ✓

Common Mistakes

Avoid these frequent errors

Using hypotenuse in area calculation
Don't use the hypotenuse (13) as base or height = wrong area calculation! The hypotenuse is never perpendicular to another side. Always identify the two perpendicular sides (legs) and use those for base and height.

Practice Quiz

Test your knowledge with interactive questions

Complete the sentence:

To find the area of a right triangle, one must multiply ________________ by each other and divide by 2.

FAQ

Everything you need to know about this question

How do I know which sides to use for base and height?

In a right triangle, use the two perpendicular sides (the legs) that meet at the 90° angle. Never use the hypotenuse, which is the longest side opposite the right angle.

Why don't all three triangles have the same area if they're all 5-12-13 triangles?

While all three triangles have sides of lengths 5, 12, and 13, they're oriented differently. The area depends on which sides are perpendicular (base and height), not just the side lengths.

Can I use any side as the base?

Only in right triangles! You can choose either leg as the base, but then the height must be the other leg. The formula $A = \frac{1}{2} \times base \times height$ requires perpendicular measurements.

How do I identify the right angle in each triangle?

Look for the square symbol in the corner of each triangle in the diagram. This marks the 90° angle where the two perpendicular sides (legs) meet.

What if I get different areas using different bases and heights?

If you're using the correct perpendicular sides, you should get the same area. Different results mean you're either using the hypotenuse incorrectly or the triangle isn't a right triangle.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Triangle questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime