Given an isosceles triangle ABC with AB = BC, let AH be the altitude from A and let D be a point on AB. Let DE be perpendicular to AC with E lying on AC, and let M be the midpoint of DE. If the distance from M to AB is 8 units and the distance from M to BC is 20 units, determine the length of AH.

Concept | Definition | Theorem |
---|---|---|

Triangle | A polygon with three sides and three angles. | The sum of the angles in a triangle is 180 degrees. |

Isosceles triangle | An isosceles triangle is a triangle that has two sides of equal length. | In an isosceles triangle, the altitudes relative to congruent sides are congruent. |

Midpoint | The point that divides a line segment into two equal parts. | The midpoint theorem for triangles, also known as the mid-segment theorem, states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half of its length. |

Perpendicular lines | Two lines or line segments that intersect at a right angle. | Two lines perpendicular to a same line are parallel to each other. |

Parallel lines | Two lines in a plane that do not intersect. | If two parallel lines are cut by a transversal, then the alternate interior angles are congruent, the corresponding angles are congruent, and the consecutive interior angles are supplementary. |

Rectangle | A rectangle is a two-dimensional shape with four straight sides and four right angles (90-degree angles). | In a rectangle, opposite sides are congruent and parallel. |

Congruence | Two triangles are said to be congruent if all corresponding sides and angles of one triangle are equal to the corresponding sides and angles of the other triangle.. | There are several ways to prove that two triangles are congruent, including the Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), and Hypotenuse-Leg (HL) criteria. |

The Geometry of Symmetry: A Poem on Isosceles Triangles and Altitudes

In a triangle, isosceles and bright

An altitude
soars in dazzling flight

Perpendicular, a line so
true

From a point on base, it splits in two

Midpoint of this line, a beacon bright

With
perfect symmetry, it shines with might

Distances from
this midpoint, oh so clear

To the sides, a measure
far and near

With these pieces laid before your eyes

The length
of the altitude you'll devise

A puzzle to solve with
care and thought

A triumph to gain, a lesson taught

In geometry, the beauty we find

A poetic dance of
the heart and mind

A world of shapes and lines so
true

A tapestry woven with a vibrant hue.

Geometry Problems

Open Problems

Visual Index

All Problems

Triangle

Isosceles Triangle

Altitude

Midpoint

Perpendicular lines

Parallel lines

Rectangle

Congruence

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