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

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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

00:14 Equal sides in an isosceles triangle

00:17 AD is perpendicular to BC according to the given data

00:21 A perpendicular in an isosceles triangle is also a median

00:26 Therefore BD equals DC equals 10

00:36 The base BC equals the sum of its parts (BD+DC)

00:43 Substitute in the value of side BC

00:50 The given ratio between side AD and BC

00:54 Substitute in the relevant values

01:06 Divide 1.2 into factors (1) and (0.2) and proceed to solve

01:15 This is the height AD

01:23 Calculate the area of the triangle ABC

01:26 Apply the formula for calculating the area of a triangle

01:29 (Height(AD) x base(BC)) divided by 2

01:33 Substitute in the relevant values according to our calculation

01:37 Calculate and solve

01:44 This is 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

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

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!

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