As shown in the figure, in the triangle ABC, ab = 13, BC = 14, AC = 15, find the length of the high ad on the side of BC  

As shown in the figure, in the triangle ABC, ab = 13, BC = 14, AC = 15, find the length of the high ad on the side of BC  

According to Pythagorean theorem
BD^2+AD^2=AB^2
DC^2+AD^2=AC^2
The two formulas subtract from each other
（BD+DC)(BD-DC)=AB^2-AC^2=13^2-15^2=-56
BD+DC=BC=14
BD-DC=-56÷14=-4
So BD = 5, DC = 9
AD^2=13^2-5^2=18×8=12^2
AD=12

In the triangle ABC, ab = 13, BC = 14, AC = 15, find the height of height ad on the side of BC
Using Pythagorean theorem

Let BD length be x, then CD length be (14-x), ad ^ 2 = 13 ^ 2-x ^ 2 = 169-x ^ 2
∵AD⊥BC
The ∧ abd and ∧ ACD are right triangles
｜ ad ^ 2 + BD ^ 2 = AB ^ 2 (Pythagorean theorem)
Ad ^ 2 + CD ^ 2 = AC ^ 2 (Pythagorean theorem)
The results are as follows
AD^2=AB^2-BD^2 ③
AD^2=AC^2-CD^2 ④
Substitute (4) for (3)
AB^2-BD^2=AC^2-CD^2
∴13^2-x^2=15^2-(14-x)^2
169-x^2=225-196+28x-x^2
169-225+196=28x
28x=140
X=5
∴AD^2=169-5^2
=169-25
=144
∴AD=12

In the triangle ABC, ab = 15, AC = 13, BC = 14, ad is perpendicular to BC to find the length of AD

Let DC = X
Then the Pythagorean theorem in right triangle ADC is: 13 ^ - x ^ = ad
In the right triangle ADB, the Pythagorean theorem is: 15 ^ - (14-x) ^ = ad
The equation: 13 ^ - x ^ = 15 ^ - (14-x)^
The solution is: x = 5
So ad = root sign (13 ^ - 5 ^) = 12

In the triangle ABC, the angle c = 90 °, CD is perpendicular to D, ab = 12, AC + BC = 17, find the length of CD

Let the length of AC be x, then the length of BC is 17-x. because the angle c = 90 °, AC + BC = AB, then x + (17-x) = 12, the value of X is solved, and then the value of X is determined according to a + b > C. finally, the length of CD is obtained according to the area method
Remember to adopt it