# Geometry Problem 1564: Find the Area of Quadrilateral BGDJ in a Right Triangle involving the
Altitude, Angle bisectors, and Midpoints.

In right triangle ABC (right angle at B), BD bisects angle ABC (BD = 8)
and BH is the altitude to AC. Bisectors of angles AHB and BHC meet AB
and BC at E and F respectively. G and J are midpoints of HE and HF. Find
the area of quadrilateral BGDJ.

Right triangle split,

Bisectors dance, midpoints meet,

Find BGDJ's shade.

####
Academic Levels: Suitable for High School and College Mathematics
Education

## Key Definitions and Descriptions

Vocabulary |
Description |

Right Triangle ABC |
A triangle with a 90-degree angle at vertex B. |

Bisector (BD) |
A line segment that divides an angle into two congruent angles. In this case, BD divides angle ABC in half. |

Altitude (BH) |
A line segment drawn from a vertex (B) perpendicular to the opposite side (AC) in a triangle. |

Right triangles (AHB & BHC) |
The two smaller right triangles formed after drawing altitude BH. AHB has a right angle at H, and BHC has a right angle at H. |

Bisectors (of AHB & BHC) |
Line segments in triangles AHB and BHC that each divide their respective angles in half and intersect the opposite side. |

Points (E & F) |
The intersection points of the bisectors from AHB and BHC with sides AB and BC, respectively. |

Midpoints (G & J) |
The middle points of segments HE and HF, respectively. |

Quadrilateral BGDJ |
The four-sided figure formed by connecting points B, G, D, and J. |

##
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Angles

Triangle

Right Triangle

Angle Bisector

Perpendicular lines

Midpoint

Altitude

Angle
of 45 degrees

Area

Quadrilateral

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