Answer :
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 [tex] \Delta ABN [/tex],
by Pythagoras theorem, we get
[tex] (Hypotenuse)^{2}=(Base)^{2}+(Perpendicular)^{2} [/tex]
[tex] (AB)^{2}=(BN)^{2}+(AN)^{2} [/tex]
[tex] (20)^{2}=(BN)^{2}+(12)^{2} [/tex]
[tex] 400=(BN)^{2}+144 [/tex]
[tex] 400-144=(BN)^{2} [/tex]
[tex] (BN)^{2}=256 [/tex]
So, BN = 16
Consider [tex] \Delta ANC [/tex],
by Pythagoras theorem, we get
[tex] (Hypotenuse)^{2}=(Base)^{2}+(Perpendicular)^{2} [/tex]
[tex] (AC)^{2}=(NC)^{2}+(AN)^{2} [/tex]
[tex] (15)^{2}=(NC)^{2}+(12)^{2} [/tex]
[tex] 225=(NC)^{2}+144 [/tex]
[tex] 225-144=(NC)^{2} [/tex]
[tex] (NC)^{2}=81 [/tex]
So, NC = 9
So, BC = BN + NC
BC = 16+9 = 25
Now consider triangle ABC,
Consider [tex] (BC)^{2}=(AB)^{2}+(AC)^{2} [/tex]
[tex] (25)^{2}=(20)^{2}+(15)^{2} [/tex]
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.
If the measure of the length BC is 25 cm. Then the triangle ΔABC will be a right-angle triangle.
What is a right-angle triangle?
It is a type of triangle in which one angle is 90 degrees and it follows the Pythagoras theorem and we can use the trigonometry function. The Pythagoras is the sum of the square of two sides is equal to the square of the longest side.
In ΔABC,AB = 20 cm, AC = 15 cm. The length of the altitude AN is 12 cm.
If the Pythagoras theorem is satisfied, then the triangle will be a right-angle triangle.
In ΔANC, we have
NC² = AC² - AN²
NC² = 15² - 12²
NC² = 225 - 144
NC² = 81
NC = 9 cm
Then in ΔANB, we have
NB² = AB² - AN²
NB² = 20² - 12²
NB² = 400 - 144
NB² = 256
NB = 16 cm
Then the value of BC will be
BC = BN + NC
BC = 16 + 9
BC = 25 cm
Then in ΔABC, we have
BC² = AB² + AC²
BC² = 20² + 15²
BC² = 400 + 225
BC² = 625
BC = 25 cm
Hence, the triangle ΔABC will be a right-angle triangle.
More about the right-angle triangle link is given below.
https://brainly.com/question/3770177
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