Triangle ABC: Finding AB Length When AD=6, CE=3, and CB=5

Q: How do I know which segments are heights in the diagram?

Look for perpendicular symbols (small squares) in the diagram! In this problem, AD is perpendicular to BC, and CE is perpendicular to AB, making them both valid heights.

Q: Why can I use two different area formulas for the same triangle?

Every triangle has multiple base-height pairs, but they all give the same area! This is why we can use both $ \frac{AD \times CB}{2} $ and $ \frac{CE \times AB}{2} $.

Q: How do I solve the equation 15 = 3AB/2?

Multiply both sides by 2 to get $ 30 = 3AB $, then divide both sides by 3 to get $ AB = 10 $. Always work step-by-step!

Triangle Area with Perpendicular Heights

Given the triangle ABC

AD=6 CE=3 CB=5

What should be the length of AB so that the area of a triangle ABC is compatible with the rest of the data in the drawing?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine the length of AB so that the triangle areas match

00:03 Examine the size of sides and lines according to the given data

00:12 They are perpendicular according to the given data

00:17 AD is also the height in the triangle

00:20 Apply the formula for calculating the area of a triangle

00:25 (height x base) divided by 2

00:38 Substitute in the relevant values and solve for the area

00:51 This is the triangle's area

01:01 Calculate the triangle's area using height and the other side

01:07 Substitute in the relevant values and proceed to solve for AB

01:27 Multiply by the denominator in order to eliminate the fraction

01:32 Isolate AB

01:42 Simplify wherever possible

01:48 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the triangle ABC

AD=6 CE=3 CB=5

What should be the length of AB so that the area of a triangle ABC is compatible with the rest of the data in the drawing?

Step-by-step solution

Given that AD is perpendicular to CB

We can establish that AD is the height of the triangle ADB

Hence the formula for the area of triangle ABC=

$\frac{AD\times CB}{2}$

We insert the existing data into the formula:

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

Due to the fact that CE is also a height, we can calculate the area of triangle ABC as follows:

$\frac{CE\times AB}{2}$

Since we found the area of triangle ABC, we will insert the data into the formula:

$15=\frac{3\times AB}{2}$

We then multiply across:

$30=3AB$

Lastly we divide both sides by 3:

$\frac{30}{3}=\frac{3AB}{3}$

$AB=10$

Final Answer

10 cm

Key Points to Remember

Essential concepts to master this topic

Area Formula: Use $\frac{base \times height}{2}$ with perpendicular segments
Method: Calculate area using AD = 6, CB = 5: $\frac{6 \times 5}{2} = 15$
Verification: Check with CE height: $\frac{3 \times 10}{2} = 15$ ✓

Common Mistakes

Avoid these frequent errors

Using non-perpendicular segments as heights
Don't use any line segment as height = wrong area calculation! Heights must be perpendicular to the base, not just any line from vertex to opposite side. Always verify that segments like AD and CE are perpendicular to their respective bases.

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 segments are heights in the diagram?

Look for perpendicular symbols (small squares) in the diagram! In this problem, AD is perpendicular to BC, and CE is perpendicular to AB, making them both valid heights.

Why can I use two different area formulas for the same triangle?

Every triangle has multiple base-height pairs, but they all give the same area! This is why we can use both $\frac{AD \times CB}{2}$ and $\frac{CE \times AB}{2}$ .

What if I get different areas using different height-base pairs?

If you get different areas, you made an error! Check that you're using perpendicular heights and the correct base measurements. All valid area calculations must give the same result.

Can I solve this without finding the area first?

No! You need the area to connect the two different base-height relationships. First find area using known values (AD and CB), then use that area with CE to find AB.

How do I solve the equation 15 = 3AB/2?

Multiply both sides by 2 to get $30 = 3AB$ , then divide both sides by 3 to get $AB = 10$ . Always work step-by-step!

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