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# Geometry

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We have a triangle \$\triangle ABC\$ and a point \$K\$ on \$BC\$ such that \$AK\$ is an altitude to \$\triangle ABC\$. If \$AC = 8,\$ \$BK = 2\$, and \$CK = 3,\$ then what is \$AB\$?

Apr 18, 2024

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Sure! We can find the length AB using the Pythagorean Theorem applied to right triangles ABK and ACK as shown in the diagram below.

[asy] pair A, B, C, K; A = (0,0); B = (0,8); C = extension(A,A+(8,0),B,B+(3,0),false); K = foot(C,A,B); draw(A--B--C--cycle); draw(A--K); label("A",A,SW); label("B",B,NW); label("C",C,SE); label("K",K,S); label("8",(A+C)/2,W); label("2",(B+K)/2,W); label("3",(C+K)/2,E); label("AB",(A+B)/2,NE); [/asy]

Theorem: The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Formulas:

AB2=AC2+BK2 (because △ABK is a right triangle)

AK2=AC2−CK2 (because △ACK is a right triangle)

Calculations:

AB2=82+22=68

Since AK is the altitude, it splits the base BC into segments BK and CK, so AB is the hypotenuse of △ABK.

AB=AB2​=68​=8.25 (round to two decimal places)

Answer: The length of side AB is 8.25 units.

Apr 18, 2024