Isosceles Triangle Area: Height 20% Greater than Base with DC=10

Q: Why is BC equal to 20 if DC is only 10?

In an isosceles triangle, the height from the vertex angle to the base always bisects the base. This means BD = DC, so BC = BD + DC = 10 + 10 = 20.

Q: How do I convert 'greater by 20%' into a calculation?

'Greater by 20%' means the new value is 100% + 20% = 120% of the original. Convert to decimal: $ \frac{120}{100} = 1.2 $, so AD = 1.2 × BC.

Q: Why can't I just add 20 to BC instead of multiplying by 1.2?

Adding 20 gives you 20 units more, not 20% more! Percentage increases are multiplicative: 20% of 20 is 4, so AD = 20 + 4 = 24, which equals 1.2 × 20.

Q: How do I know which formula to use for triangle area?

For any triangle, use Area = $ \frac{1}{2} \times \text{base} \times \text{height} $. Here, BC is the base (20) and AD is the height (24), giving us $ \frac{20 \times 24}{2} = 240 $.

Isosceles Triangle Properties with Percentage Relationships

Triangle ABC is an isosceles triangle AB=AC

AD is the height of the BC

Given DC=10

The length of the height AD is greater by 20% than the length of the side BC.

Calculate the area of the triangle ABC

❤️ 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

00:16 Let's calculate the area of triangle ABC.

00:30 An Isosceles triangle has two equal sides.

00:35 Line AD is perpendicular to line BC as shown.

00:39 In an Isosceles triangle, the perpendicular is also a median.

00:45 This means BD equals DC, and both are 10.

00:52 The base, BC, is BD plus DC.

00:59 Let's substitute the value for side BC.

01:06 We know the ratio between AD and BC.

01:10 Now, substitute the values needed.

01:22 Divide one point two into one and zero point two, then solve.

01:31 This gives us the height, AD.

01:39 Back to calculating the area of triangle ABC.

01:44 Use the formula: Area equals height times base divided by two.

01:49 Let's plug in the numbers we found.

01:52 Now, let's calculate and solve.

01:55 And there you have it, the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Triangle ABC is an isosceles triangle AB=AC

AD is the height of the BC

Given DC=10

The length of the height AD is greater by 20% than the length of the side BC.

Calculate the area of the triangle ABC

Step-by-step solution

Given that it is an isosceles triangle if $DC=10$ then $BC=20$ .

We are told that the height $AD$ is greater by $20%$ percent

than the length of the side $BC$ .

That is:

$AD=1.2\cdot BC$

$\frac{100}{100}+\frac{20}{100}=\frac{120}{100}=1.2$

$AD=1.2\cdot20=24$

From here we can calculate the area of the triangle $ΔABC$ :

$AΔ\text{ABC}=\frac{AD\cdot BC}{2}=\frac{24\cdot20}{2}=\frac{480}{2}=240$

Final Answer

240 cm²

Key Points to Remember

Essential concepts to master this topic

Isosceles Property: Height AD bisects base BC making BD = DC
Percentage Technique: Height AD = 1.2 × BC where 120% = 1.2
Area Check: Substitute values: $\frac{24 \times 20}{2} = 240$ ✓

Common Mistakes

Avoid these frequent errors

Confusing DC with BC in percentage calculations
Don't use DC = 10 directly in the percentage calculation = wrong base length! This gives height = 12 instead of 24. Always recognize that in isosceles triangles, D bisects BC, so BC = 2 × DC.

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 is BC equal to 20 if DC is only 10?

In an isosceles triangle, the height from the vertex angle to the base always bisects the base. This means BD = DC, so BC = BD + DC = 10 + 10 = 20.

How do I convert 'greater by 20%' into a calculation?

'Greater by 20%' means the new value is 100% + 20% = 120% of the original. Convert to decimal: $\frac{120}{100} = 1.2$ , so AD = 1.2 × BC.

Why can't I just add 20 to BC instead of multiplying by 1.2?

Adding 20 gives you 20 units more, not 20% more! Percentage increases are multiplicative: 20% of 20 is 4, so AD = 20 + 4 = 24, which equals 1.2 × 20.

How do I know which formula to use for triangle area?

For any triangle, use Area = $\frac{1}{2} \times \text{base} \times \text{height}$ . Here, BC is the base (20) and AD is the height (24), giving us $\frac{20 \times 24}{2} = 240$ .

What if I calculated the height wrong?

Double-check by working backwards: if Area = 240 and BC = 20, then height = $\frac{2 \times 240}{20} = 24$ . Verify: is 24 exactly 20% more than 20? Yes, because 24 = 1.2 × 20!

🌟 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