Calculate Triangle Area: Finding Area with Sides 13cm, 15cm, and Height from Point D

Triangle Area with Perimeter Constraints

The triangle ABC has a perimeter measuring 42 cm.

AD = 12

AC = 15

AB = 13

Calculate the area of the triangle.

131313151515121212AAABBBCCCDDD

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Calculate the area of the triangle ABC
00:04 Substitute in the relevant values according to the data
00:13 The perimeter of the triangle equals the sum of the sides
00:23 Substitute in the relevant values according to the data
00:34 Calculate and solve to find BC
00:47 Isolate BC
00:58 This is the base length BC
01:02 Calculate the area of the triangle ABC
01:05 Apply the formula for calculating the area of the triangle
01:08 (height(AD) x base (BC)) divided by 2
01:12 Substitute in the relevant values
01:16 Calculate and solve
01:25 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

The triangle ABC has a perimeter measuring 42 cm.

AD = 12

AC = 15

AB = 13

Calculate the area of the triangle.

131313151515121212AAABBBCCCDDD

2

Step-by-step solution

Given that the perimeter of triangle ABC is 42.

We will use this data to find side CB:

13+15+CB=42 13+15+CB=42

CB+28=42 CB+28=42

CB=4228=14 CB=42-28=14

Now we can calculate the area of triangle ABC:

AD×BC2=12×142=1682=84 \frac{AD\times BC}{2}=\frac{12\times14}{2}=\frac{168}{2}=84

3

Final Answer

84 cm²

Key Points to Remember

Essential concepts to master this topic
  • Perimeter Rule: Sum of all three sides equals total perimeter
  • Height Formula: Area = 12×base×height \frac{1}{2} \times base \times height = 12×142=84 \frac{12 \times 14}{2} = 84
  • Check: Verify BC = 42 - 13 - 15 = 14 cm ✓

Common Mistakes

Avoid these frequent errors
  • Using the wrong base-height pair
    Don't use AB = 13 as the base with height AD = 12! This gives wrong area since AD isn't perpendicular to AB. The height AD = 12 is perpendicular to base BC, so use BC = 14. Always identify which side the height is perpendicular to.

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

How do I know which side is the base when calculating area?

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The base is the side that the height is perpendicular to. In this problem, height AD = 12 is perpendicular to side BC, so BC is your base!

Why can't I just use any two sides I know?

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Triangle area requires a base and its corresponding height (perpendicular distance). You can't randomly pick two sides - the height must form a 90° angle with the base.

What if I forgot to find the third side first?

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You must find all three sides using the perimeter! Without knowing BC = 14, you can't calculate the area since BC is the base that corresponds to height AD = 12.

How do I verify my area calculation is correct?

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Check your work by ensuring: 1) All three sides add up to the given perimeter, 2) You used the correct base-height pair, and 3) Your arithmetic is accurate.

Can I use a different base-height combination?

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Yes, but you'd need the height to a different base! This problem gives you height AD = 12 to base BC, so that's the most direct approach.

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