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Lynna [10]
3 years ago
14

In ΔABC,AB = 20 cm, AC = 15 cm. The length of the altitude AN is 12 cm. Prove that ΔABC is a right triangle.

Mathematics
2 answers:
RSB [31]3 years ago
6 0

Refer to the attached image.

Since AN is an altitude, an altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the base forming a right angle with the base.

Consider \Delta ABN,

by Pythagoras theorem, we get

(Hypotenuse)^{2}=(Base)^{2}+(Perpendicular)^{2}

(AB)^{2}=(BN)^{2}+(AN)^{2}

(20)^{2}=(BN)^{2}+(12)^{2}

400=(BN)^{2}+144

400-144=(BN)^{2}

(BN)^{2}=256

So, BN = 16

Consider \Delta ANC,

by Pythagoras theorem, we get

(Hypotenuse)^{2}=(Base)^{2}+(Perpendicular)^{2}

(AC)^{2}=(NC)^{2}+(AN)^{2}

(15)^{2}=(NC)^{2}+(12)^{2}

225=(NC)^{2}+144

225-144=(NC)^{2}

(NC)^{2}=81

So, NC = 9

So, BC = BN + NC

BC = 16+9 = 25

Now consider triangle ABC,

Consider (BC)^{2}=(AB)^{2}+(AC)^{2}

(25)^{2}=(20)^{2}+(15)^{2}

625 = 400 + 225

625 = 625

Therefore, by the converse of Pythagoras theorem , which states that "If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle".

Therefore, triangle ABC is a right triangle.

Makovka662 [10]3 years ago
4 0
It is the pythagorean theorem converse
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