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AB = 30 in and BC = 50 in.

We use Pythagorean theorem to solve this.  Since AN is an altitude, this means that it is perpendicular to BC.  This means BN and AN are the legs of one right triangle, with AB being the hypotenuse:

18²+24² = AB²
324 + 576 = AB²
900 = AB²

Take the square root of both sides:
√900 = √AB²
30 = AB

NC and AN form the legs of the other right triangle, with AC being the hypotenuse:

24²+NC² = 40²
576 + NC² = 1600

Subtract 576 from both sides:
576 + NC² - 576 = 1600 - 576
NC² = 1024

Take the square root of both sides:
√NC² = √1024
NC = 32

BC = BN + NC = 18 + 32 = 50

The value of AB and BC will be 30 in and 50 in. Pythagoras theorem is used to solve the problem.

What is trigonometry?

The field of mathematics is concerned with the relationships between triangles' sides and angles, as well as the related functions of any angle.

The given data in the problem is;

The altitude is, AN = 24 in

BN = 18 in

From the pythogorous theorem;

[tex]\rm h^2 = p^2+b^2 \\\\ BN^2+AN^2=AB^2 \\\\ 324+576=AB^2 \\\\ AB=\sqrt{900} \\\\ AB=30 \ inch[/tex]

Pythorous theorem applied in the other triangle;

[tex]\rm AN^2 +NC^2 = AC^2 \\\\ \rm 24^2 +NC^2 = 40^2 \\\\ 576+NC^2=1600 \\\\ NC^2 = 1600-576 \\\\ NC^2=1024 \\\\ NC=\sqrt{1024} \\\\ NC=32[/tex]

The side BC is the sum of the  side BN and NC will be;

[tex]\rm BC = BN+NC \\\\ BC=18+32 \\\\ BC=50\ inch[/tex]

Hence the value of AB and BC will be 30 inches and 50 inches.

To learn more about the trigonometry refer to the link;

https://brainly.com/question/26719838

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