Examkin

Soru

Zorluk: Çok zorPythagorean Theorem and Special Right Triangles

A rectangular piece of paper ABCDABCD has dimensions AB=12AB = 12 inches and BC=9BC = 9 inches. The paper is folded so that vertex AA falls directly on vertex CC, creating a crease EFEF where EE lies on ABAB and FF lies on CDCD. What is the length of the crease EFEF, in inches?

Cevap: 11.25 inches

Cevap

The length of the crease is 11.25 inches.
By interpreting the fold geometrically, we establish that the crease is the perpendicular bisector of the diagonal. We can solve for the segments along the side using a right triangle formed at the corner, and then construct a second right triangle using an altitude to find the length of the crease hypotenuse.

Adım Adım Çözüm

1
Identify the relationship created by the fold.
The crease EFEF is the perpendicular bisector of the diagonal ACAC, meaning AE=EC=xAE = EC = x.
When a point is folded onto another, the crease line acts as the perpendicular bisector of the segment connecting the two points.
2
Set up an expression for the remaining part of the side ABAB.
Since AB=12AB = 12 and AE=xAE = x, the length of EB=12−xEB = 12 - x.
The point EE lies on segment ABAB, dividing it into AEAE and EBEB.
3
Use the Pythagorean Theorem in right triangle EBCEBC to solve for xx.
x2=(12−x)2+92  ⟹  x2=144−24x+x2+81  ⟹  24x=225  ⟹  x=9.375x^2 = (12 - x)^2 + 9^2 \implies x^2 = 144 - 24x + x^2 + 81 \implies 24x = 225 \implies x = 9.375.
The triangle EBCEBC is a right triangle with legs EBEB and BCBC, and hypotenuse ECEC.
4
Form a second right triangle to find the length of the crease EFEF.
Draw FG⊥ABFG \perp AB with GG on ABAB. This forms right triangle EGFEGF with legs FG=9FG = 9 and EG=AB−EB−DF=12−2.625−2.625=6.75EG = AB - EB - DF = 12 - 2.625 - 2.625 = 6.75.
Constructing an altitude from FF to ABAB allows us to create a right triangle that has the crease EFEF as its hypotenuse.
5
Apply the Pythagorean Theorem to right triangle EGFEGF to calculate the final length of EFEF.
EF=6.752+92=45.5625+81=126.5625=11.25EF = \sqrt{6.75^2 + 9^2} = \sqrt{45.5625 + 81} = \sqrt{126.5625} = 11.25 inches.
The hypotenuse of right triangle EGFEGF represents the length of the crease.

Anahtar Kavram

Applying the Pythagorean Theorem to geometric folds and multi-step right triangle relationships
Bu soruyu puanla

Konular

Pythagorean Theorem and Special Right Triangles

Bu konudan daha fazla soru