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Home » Triangles » Isosceles Triangles » Isosceles Triangles: the Median to the Base is Perpendicular to the Base

Isosceles Triangles: the Median to the Base is Perpendicular to the Base

Last updated: Oct 1, 2019 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

In a triangle, a line that connects one corner (or vertice) to the middle point of the opposite side is called a median. A property of isosceles triangles, which is simple to prove using triangle congruence, is that in an isosceles triangle the median to the base is perpendicular to the base.

Problem

Show that in an isosceles triangle ΔABC, the median to the base, AD,  is perpendicular to the base.

isosceles triangle with median

Strategy

We will again work backward from what we need to ultimately show. To prove that two lines are perpendicular to each other, we need to show that the angle between them is 90°.

One tool that helps us show that an angle is 90° is the Linear Pair Perpendicular Theorem - if two straight lines intersect at a point and form a linear pair of equal angles, they are perpendicular.

This is great, since triangle congruence can show that angles are equal. And here, proving that triangles are congruent is almost too easy! All the sides are given as equal, so the triangles are congruent using the Side-Side-Side postulate.

Proof

(1) ΔABC is isosceles                          //Given

(2) AD is the median to the base, AB  //Given

(3) AD=AD                                           //Common side

(4) AB=AC                                           //(1), definition of isosceles triangle

(5) BD=DC                                           //(2), definition of median

(6) △ABD≅△ACD                               // Side-Side-Side postulate.

(7) ∠ADB ≅ ∠ADC                              // Corresponding angles in congruent triangles (CPCTC)

(8) AD⊥BC                                          // Linear Pair Perpendicular Theorem

« Diagonals of Rectangles are of Equal Length
The Diagonals of a Parallelogram Bisect Each Other »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected].

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About

Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

Geometry Topics

  • Area of Geometric Shapes
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    • Similar Triangles
    • Triangle Inequalities

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