How to find the area of a square inscribed in a right triangle?

Geometry Level 3

Triangle ABC is a right triangle with the 90 degree angle located on Angle A. Point W is a point on segment AB, Point X on segment AC, and Points Y and Z on Segment BC such that WXYZ is a square.

If BZ = 6 and YC = 8 (the measure of the side of the square is the missing measure in segment BC), find the area of the square WXYZ.


The answer is 48.

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Emmanuel John Baliwag by Emmanuel John Baliwag , 21, Philippines

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1 solution

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Aug 4, 2014

BZ=6 YC=8 B Z W Z = X Y Y C \frac{BZ}{WZ}=\frac{XY}{YC} ⇒ W Z 2 = 48 \Rightarrow WZ^2=\boxed{48}

7 Helpful 0 Interesting 0 Brilliant 0 Confused

It appears I have been confused by the triangles' orientations. It is clear now. THanks everyone! :D

Emmanuel John Baliwag - 6 years, 10 months ago

Hello there. Thank you for including the solution. I am just beginning to be advanced in this kinds of questions but may I ask? How did you kow that Triangles BZW and YCX are proportional, despite the two of them having measures of 6, x and 8, x respectively? Thank you!

Emmanuel John Baliwag - 6 years, 10 months ago

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Use the AAA similarity criterion: If the corresponding angles of two triangles are equal, then their corresponding sides are proportional and the triangles are similar.

Enoc Cetina - 6 years, 10 months ago

angle BWZ + angle CXY = angle CAB = 90 degrees

therefore angle BWZ = 90 degrees - angle CXY

however as angle BWZ = 90 degrees - angle WBZ

angle CXY = angle WBZ

angle CYX = angle WZB = 90 degrees

angle XCY = angle BWZ

David Farrar - 6 years, 10 months ago

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