Nested Triangles
Part of the Triangles section of Coddy's Geometry journey. Lesson 51 of 63.
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.
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 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
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Triangles
1Sides and Names
Three Sides, Three CornersSorting by SidesNaming by AnglesTicks into EquationsThe Longest SideRecap: Sorting Triangles4Area of a Triangle
Half the RectangleHeight Is Not a SideRight Triangle AreaFinding the HeightAny Side Can Be the BaseRecap: Area10Scale in the World
ShadowsAcross the RiverNested TrianglesA Side with an ExpressionRecap: Scale in the World2The Triangle Inequality
Can Three Sticks Meet?Exactly ReachingThe Range of the Third SideThe FormulaRecap: The Third Side5The Pythagorean Theorem
Squares on the SidesFinding the HypotenusePythagorean TriplesWhen It Is Not WholeRecap: The Hypotenuse3Perimeter
Around the EdgeA Missing SideIsosceles PerimeterSides as ExpressionsRecap: Around the Edge6Finding a Leg
Working BackwardsThe LadderWhich Side Is c?A Leg That Is Not WholeRecap: Finding a Leg