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Nested Triangles

Part of the Triangles section of Coddy's Geometry journey. Lesson 48 of 60.

Draw a line across a triangle parallel to one side, and it cuts off a smaller triangle similar to the whole. The corner at the top is shared, and the parallel line makes corresponding angles equal at the base, so AA holds.

$Triangle ABC cut by DE parallel to BC$

Here DE is parallel to BC, so △ADE ~ △ABC, with D matching B. Careful: the side that matches AD is the whole of AB, not just DB. AB = 3 + 6 = 9, so the factor is $\frac{9}{3} = 3$ , and BC = 3 × 4 = 12.

That makes it a two-step chase: first the whole side with Segment addition, then the ratio with Similar triangles.

Challenge

In △ABC the segment DE is parallel to BC (the chevrons). AD = 4, DB = 8 and DE = 5. Find BC, then press Check.

Try it yourself

$Find the marked part of the figure$

Find the marked part of the figure

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Triangles

1The Triangle Inequality

Can Three Sticks Meet?Exactly Reaching The Range of the Third Side Recap: The Third Side

4Area of a Triangle

Half the Rectangle Height Is Not a Side Right Triangle Area Finding the Height Any Side Can Be the Base Recap: Area

7Is It Right?

The Converse Testing a Triangle Acute or Obtuse?Sorting by the Test Recap: Right or Not

10Scale in the World

Shadows Across the RiverNested TrianglesA Side with an Expression Recap: Scale in the World

2Sides and Names

Sorting by Sides The Longest Side Ticks into Equations Recap: Sorting Triangles

5The Pythagorean Theorem

Squares on the Sides Finding the Hypotenuse Pythagorean Triples When It Is Not Whole Recap: The Hypotenuse

8Special Right Triangles

The 45-45-90 Triangle Legs from the Hypotenuse The 30-60-90 Triangle All Three Sides Which Triangle Is It?Recap: Special Triangles

11Two Tools at Once

Height of an Isosceles Area from the Height The Equilateral Triangle Right Triangle Perimeter Recap: Two Tools