Calculate the Median Length AD in an Isosceles Triangle with Area 60 cm²

Q: Why is BC = 10 cm if BD = 5 cm?

Since AD is the median, point D is the midpoint of BC. This means BD = DC = 5 cm, so the total base BC = 5 + 5 = 10 cm.

Q: Can I use a different formula for the area?

Yes! You could use $ Area = \frac{1}{2}ab\sin C $, but since AD is perpendicular to BC in an isosceles triangle, the base × height formula is much simpler here.

Q: How do I know AD is the height of the triangle?

In an isosceles triangle, the median from the apex (vertex angle) to the base is also the altitude. This means AD ⊥ BC, making AD the height.

Q: Why do we solve 60 = 5 × AD instead of using another method?

This comes directly from the area formula! We know $ 60 = \frac{1}{2} \times 10 \times AD $, which simplifies to $ 60 = 5 \times AD $. Simple algebra gives us the answer.

Area Formula with Median Properties

Triangle ABC is isosceles.

AD is the median of the BC.
ABC has an area of 60 cm².
BD = 5

Calculate the length AD.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate AD

00:04 We'll identify the sides and lines according to the given data

00:11 Equal sides according to the given data (isosceles)

00:15 Perpendicular according to the given data

00:24 In an isosceles triangle, the height is also a median

00:39 The whole side equals the sum of its parts

00:49 Apply the formula for calculating the triangle's area

00:55 (height x base) divided by 2

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

01:08 Multiply by denominators to eliminate fractions

01:16 Isolate AD

01:33 This is the solution

Step-by-step written solution

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Understand the problem

Triangle ABC is isosceles.

AD is the median of the BC.
ABC has an area of 60 cm².
BD = 5

Calculate the length AD.

Step-by-step solution

To calculate the length of the median AD, follow these steps:

Step 1: Calculate the length of the base BC. Since AD is the median, and BD = DC = 5 cm, the total length is $BC = 5 + 5 = 10 \, \text{cm}$ .
Step 2: Use the area formula for triangle ABC, where $S = \frac{1}{2} \times BC \times AD$ . Given $S = 60 \, \text{cm}^2$ , substitute the known values: $60 = \frac{1}{2} \times 10 \times AD$ .
Step 3: Solve for $AD$ by simplifying: $60 = 5 \times AD$ implies $AD = \frac{60}{5} = 12 \, \text{cm}$ .

Therefore, the length of the median AD is $12 \, \text{cm}$ .

Final Answer

12 cm

Key Points to Remember

Essential concepts to master this topic

Median Property: In isosceles triangles, median to base equals altitude
Area Formula: $Area = \frac{1}{2} \times base \times height = \frac{1}{2} \times 10 \times AD$
Check: Substitute back: $\frac{1}{2} \times 10 \times 12 = 60$ ✓

Common Mistakes

Avoid these frequent errors

Using BD as the full base length
Don't use BD = 5 as the base length = Area = 30! Since D is the midpoint, BD is only half the base. Always remember BC = BD + DC = 5 + 5 = 10 cm for the complete base.

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 = 10 cm if BD = 5 cm?

Since AD is the median, point D is the midpoint of BC. This means BD = DC = 5 cm, so the total base BC = 5 + 5 = 10 cm.

Can I use a different formula for the area?

Yes! You could use $Area = \frac{1}{2}ab\sin C$ , but since AD is perpendicular to BC in an isosceles triangle, the base × height formula is much simpler here.

How do I know AD is the height of the triangle?

In an isosceles triangle, the median from the apex (vertex angle) to the base is also the altitude. This means AD ⊥ BC, making AD the height.

What if the triangle wasn't isosceles?

If the triangle wasn't isosceles, AD would still be a median, but it wouldn't necessarily be perpendicular to BC. You'd need additional information to find the actual height.

Why do we solve 60 = 5 × AD instead of using another method?

This comes directly from the area formula! We know $60 = \frac{1}{2} \times 10 \times AD$ , which simplifies to $60 = 5 \times AD$ . Simple algebra gives us the answer.

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