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In the right triangle ABC, ∠ C = 90 °, BC = 8, triangle ABC area = 24, find the height on the hypotenuse AB (many methods)
Analysis:
BC = 8, triangle area s = 1 / 2 * BC * AC = 24, AC = 6
AB = 10 from Pythagorean theorem
Let the height of the hypotenuse AB be CD and the point d be perpendicular
Method 1. (area) s = 1 / 2 ab * CD = 24
So CD = 24 / 5
Method 2. (acute trigonometric function)
sin∠A＝BC/AB=CD/AC
So CD = BC * AC / AB = 24 / 5
Method 3. (similar triangle)
It is easy to know that ACD is similar to ABC
So CD / BC = AC / AB
That is CD = BC * AC / AB = 24 / 5
In △ ABC, point D is a point on the straight line BC. AB = 15, ad = 12, AC = 13, BD = 9 are known. Find the length of BC
In RT △ ADC, DC = ac2 − ad2 = 5, then BC = BD + DC = 14
Known: in the triangle ABC, angle a = 90, ad is the height of BC, ab = 4, ad = 12 / 5, find AC and BC
In RT △ ADB, BD ^ 2 = AB ^ 2-ad ^ 2 = 4 ^ 2 - (5 / 12) ^ 2, so BD = 4 / 5
It is easy to know RT △ ADC ∽ RT △ BDA, so ad ^ 2 = CD * BD, so CD = 36 / 5, then CB = 8, AC = 4 * radical 3
In the known triangle ABC, ∠ a = 90 °, ad is the height of BC, ab = 4, Ad12 / 5, find the length of AC and BC?
From triangle area: S = 1 / 2 * AB * AC = 1 / 2 * ad * BC
So AB * AC = ad * BC
Substituting AB = 4 and ad = 12 / 5, we get 4 * AC = 12 / 5 * BC
So AC = 3 / 5 * BC
Let BC = x, AC = 3 / 5 * X
From Pythagorean theorem: AC * AC + AB * AB = BC * BC
Namely (3 / 5 * x) * (3 / 5 * x) + 4 * 4 = x * x
It is reduced to X * x = 25
So x = 5 3 / 5 * x = 3
Then BC = 5, AC = 3
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