In isosceles triangle ABC, we have AB=AC=4. The altitude from B meets AC at H. If AH=2(HC) then determine BC.
Let's assume the triangle is ABC.
Given that AB = AC = 4.
Let BH be the altitude from B meeting AC at H.
Given that AH = 2(HC).
Let's assume HC = x, then AH = 2x.
By using Pythagoras theorem in triangle ABH, we have:
AB^2 = AH^2 + BH^2
4^2 = (2x)^2 + BH^2
16 = 4x^2 + BH^2
BH^2 = 16 - 4x^2 ---- (1)
Similarly, by using Pythagoras theorem in triangle AHC, we have:
AC^2 = AH^2 + HC^2
4^2 = (2x)^2 + x^2
16 = 4x^2 + x^2
16 = 5x^2
x^2 = 16/5
x = √(16/5) = 4/√5 = 4√5/5
By substituting the value of x in equation (1), we have:
BH^2 = 16 - 4(4√5/5)^2
BH^2 = 16 - 4(16/5)
BH^2 = 16 - 64/5
BH^2 = (80 - 64)/5
BH^2 = 16/5
BH = √(16/5) = 4/√5 = 4√5/5
Since BH = HC, the length of BC is equal to BH + HC.
BC = 4√5/5 + 4√5/5
BC = (4√5 + 4√5)/5
BC = (8√5)/5
BC = (8/5)√5
Therefore, the length of BC is (8/5)√5.
To determine the length of BC in the given isosceles triangle ABC, where AB = AC = 4, and the altitude from B meets AC at H, and AH = 2(HC), we can use the Pythagorean theorem and the property of isosceles triangles.
Let's denote the length of BC as x.
From the given information, we know that AH = 2(HC). This means that the height of the triangle, BH, is twice the height of the smaller triangle formed by dividing the base AC into two equal parts at H.
By applying the Pythagorean theorem to the right-angled triangle ABH, we can find the length of BH:
BH^2 = AB^2 - AH^2
BH^2 = 4^2 - (2(HC))^2
BH^2 = 16 - 4(HC)^2
BH^2 = 16 - 4(HC)(HC)
BH^2 = 16 - 4HC^2
Now, consider the right-angled triangle BHC. By applying the Pythagorean theorem to it, we can express BH^2 in terms of x and HC:
BH^2 = HC^2 + (BC/2)^2
16 - 4HC^2 = HC^2 + (x/2)^2
Since AB = AC, the triangle ABC is isosceles, and BC = x, we can write:
2HC = x/2
HC = x/4
Plugging this value back into the equation we derived from the triangle BHC, we have:
16 - 4(x/4)^2 = (x/4)^2 + (x/2)^2
16 - x^2/4 = x^2/16 + x^2/4
64 - x^2 = 4x^2 + 16x^2
64 - x^2 = 20x^2
21x^2 = 64
x^2 = 64/21
Taking the square root of both sides, we have:
x = sqrt(64/21)
Therefore, the length of BC is sqrt(64/21).