Triangle Area with 9-Unit Median-Height: Special Case Calculation

Q: What does it mean that AD is both a median and height?

A median connects a vertex to the midpoint of the opposite side, while a height is perpendicular to the base. When AD is both, point D is exactly halfway between B and C, and AD forms a 90° angle with BC.

Q: Why is the base 13 instead of 6.5?

The diagram shows BD = 6.5, but that's only half the base! Since D is the midpoint of BC, we have DC = 6.5 too. The complete base is BC = BD + DC = 6.5 + 6.5 = 13.

Q: How do I know this triangle is special?

This triangle is isosceles! When the median from vertex A is also the height, it means AB = AC. The triangle has perfect symmetry across line AD.

Q: Can I split this into two smaller triangles?

Yes! Triangle ABC splits into two identical right triangles: ABD and ACD. Each has area = ½ × 6.5 × 9 = 29.25, so total area = 29.25 + 29.25 = 58.5.

Triangle Area with Median-Height Property

AD is the median and the height of triangle ABC.

AD = 9

Calculate the area of triangle ABC.

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Step-by-step video solution

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00:00 Calculate the area of triangle ABC

00:03 AD is a median according to the given data. The median bisects the side

00:08 Insert the value of BD into the formula according to the given data, in order to determine DC

00:12 The entire side equals the sum of its parts

00:17 Substitute in the relevant values and proceed to solve to determine BC

00:22 Apply the formula for calculating a triangle area

00:26 (Base X Height) divided by 2

00:35 Substitute the appropriate values into the formula and proceed to solve it in order to determine the area

00:48 This is the solution

Step-by-step written solution

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Understand the problem

AD is the median and the height of triangle ABC.

AD = 9

Calculate the area of triangle ABC.

Step-by-step solution

To solve the problem of finding the area of triangle $ABC$ , we follow these steps:

Step 1: Recognize that $AD$ is a median and also the height, hence it bisects $BC$ into two equal lengths.
Step 2: Given $BD = DC = 6.5$ , determine $BC = 2 \times 6.5 = 13$ .
Step 3: Use the area formula for triangles, $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ .
Step 4: Substitute the known values: $\text{Area} = \frac{1}{2} \times 13 \times 9 = 58.5$ .

The area of triangle $ABC$ is, therefore, $58.5$ .

Final Answer

58.5

Key Points to Remember

Essential concepts to master this topic

Special Property: When AD is both median and height, it creates symmetrical triangles
Calculation: Area = ½ × base × height = ½ × 13 × 9 = 58.5
Verification: Check that D is midpoint: BD = DC = 6.5, so BC = 13 ✓

Common Mistakes

Avoid these frequent errors

Using only half the base length
Don't calculate Area = ½ × 6.5 × 9 = 29.25! This uses only BD instead of the full base BC. The median divides BC into two equal parts, but the area formula needs the complete base length. Always use the full base: BC = BD + DC = 6.5 + 6.5 = 13.

Practice Quiz

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Is the straight line in the figure the height of the triangle?

FAQ

Everything you need to know about this question

What does it mean that AD is both a median and height?

A median connects a vertex to the midpoint of the opposite side, while a height is perpendicular to the base. When AD is both, point D is exactly halfway between B and C, and AD forms a 90° angle with BC.

Why is the base 13 instead of 6.5?

The diagram shows BD = 6.5, but that's only half the base! Since D is the midpoint of BC, we have DC = 6.5 too. The complete base is BC = BD + DC = 6.5 + 6.5 = 13.

How do I know this triangle is special?

This triangle is isosceles! When the median from vertex A is also the height, it means AB = AC. The triangle has perfect symmetry across line AD.

Can I split this into two smaller triangles?

Yes! Triangle ABC splits into two identical right triangles: ABD and ACD. Each has area = ½ × 6.5 × 9 = 29.25, so total area = 29.25 + 29.25 = 58.5.

What if the median wasn't also the height?

Then you'd need more information! You'd need either the length of the base BC or the perpendicular height from A to BC. The special property that AD is both median and height makes this problem solvable.

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