Using the Pythagorean Theorem to Find Triangle Area

🔑 Key Concepts

• An isosceles triangle has two congruent sides.
• The height of an isosceles triangle can split the base into two equal parts.
• Drawing the height creates two right triangles inside the larger triangle.
• The Pythagorean theorem can be used to find the missing height.
• Once the height is found, use \(A=\frac{bh}{2}\) to find the area of the triangle.

✏️ Worked Example

Example: Area of an Isosceles Triangle

Problem from the video: An isosceles triangle has congruent sides of \(10\) cm. The base is \(12\) cm. What is the area of the triangle?

Since the triangle is isosceles, the height from the top vertex splits the base into two equal parts.

\(12\div2=6\)

Now use one of the right triangles. The hypotenuse is \(10\) cm, one leg is \(6\) cm, and the missing height is \(h\).

\(a^2+b^2=c^2\)

\(6^2+h^2=10^2\)

\(36+h^2=100\)

\(h^2=64\)

\(h=8\)

The height of the triangle is \(8\) cm.

Now use the area formula for a triangle.

\(A=\frac{bh}{2}\)

\(A=\frac{12(8)}{2}\)

\(A=\frac{96}{2}\)

\(A=48\)

Answer: The area is \(\boxed{48\text{ cm}^2}\).

Video Connection

This problem is really a Pythagorean theorem application. The area formula is used at the end, but the key step is drawing the height of the isosceles triangle, splitting the base into two equal pieces, and using a right triangle to find the missing height.

🧠 Math Vocabulary

• Isosceles Triangle: A triangle with two congruent sides.
• Congruent: Having the same size and shape. Congruent sides have equal lengths.
• Base: The side of a triangle used as the reference side when finding area.
• Height: The perpendicular distance from the base to the opposite vertex.
• Right Triangle: A triangle with one \(90^\circ\) angle.
• Hypotenuse: The longest side of a right triangle, located across from the right angle.
• Pythagorean Theorem: A rule for right triangles that states \(a^2+b^2=c^2\).
• Area of a Triangle: The amount of space inside a triangle, found with \(A=\frac{bh}{2}\).

💡 Main Idea

In this lesson, students use the Pythagorean theorem to find the missing height of an isosceles triangle, then use that height to calculate the triangle’s area.

📚 What You Should Already Know

Students should know how to identify an isosceles triangle, split a base into equal parts, recognize a right triangle, square numbers, find square roots, and use the area formula for a triangle.

🚀 What Comes Next

After solving this problem, students can apply the Pythagorean theorem to other geometry problems involving height, area, composite figures, diagonals, and real-world measurements.

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