In the isosceles triangle ACB AB=AC=4cm BAC= 30. and CN is an altitude. Find BN
(BC/2)/4 = sin 15°
so, BC = 8sin15°
angle BCN is also 15°
BN/BC = sin 15°
BN = 8 sin^2 15° = 4-2√3
But the textbook answer is 2(2-√3)cm
Why did the triangle bring a measuring tape to the party? Because it wanted to show off its equal sides of 4cm! Now, since the triangle is isosceles and BAC is 30 degrees, we know that BNC is also 30 degrees.
Using the cosine rule, we can find the length of BN:
BN^2 = BC^2 + CN^2 - 2(BC)(CN)cos(BNC)
But since BC = 4cm and CN is the altitude, we can rewrite the formula as:
BN^2 = 4^2 + CN^2 - 2(4)(CN)cos(30)
Now, since BNC is right-angled due to CN being the altitude, we know that cos(30) = sqrt(3)/2. Plugging that in, we have:
BN^2 = 16 + CN^2 - 4CN(sqrt(3))
But here's the kicker, in any right-angled triangle, the altitude times the hypotenuse is equal to the product of the two sides adjacent to the right angle. In this case, that means:
CN * 4 = BC * BN
4CN = 4BN
CN = BN
So, let's substitute that into our formula:
BN^2 = 16 + BN^2 - 4BN(sqrt(3))
Simplifying this quadratic equation, we have:
0 = 16 - 4BN(sqrt(3))
Solving for BN, we get:
BN = 16 / (4(sqrt(3)))
BN = 4 / sqrt(3)
Therefore, BN is approximately equal to 2.31 cm.
To find BN, we need to use the properties of an isosceles triangle and trigonometry.
Since AB = AC = 4 cm, we can conclude that triangle ABC is an equilateral triangle because all sides are equal.
In an equilateral triangle, all angles are also equal, so angle BAC is also equal to angle BCA, which is 60 degrees.
Now, let's draw the altitude CN in the triangle.
Since angle BAC is 30 degrees, we can consider right triangle ANC, where AN is the altitude.
In a right triangle, we can use trigonometric ratios to find the length of BN.
Let's consider the trigonometric ratio for the angle 30 degrees:
sin(30) = opposite / hypotenuse
In triangle ANC, the opposite side is BN, and the hypotenuse is AC, which is 4 cm.
sin(30) = BN / 4
Now, let's solve for BN:
BN = 4 * sin(30)
To calculate the value of sin(30), we can use the exact value:
sin(30) = 1/2
Therefore,
BN = 4 * 1/2
= 2 cm
So, the length of BN is 2 cm.