ABC is an isosceles triangle. AB=BC=6. Angle ABC=120,AC=a square root of b. What is a+b?
Drop an altitude from B to D on AC. Then triangles ABD and CBD are 30-60-90 right triangles.
AD = DC = 3√3
so, AC = 6√3
a+b = 9
To find the value of a + b, we'll first need to find the length of AC.
Since ABC is an isosceles triangle, AB = BC. Therefore, we can draw an altitude from angle B to side AC, splitting triangle ABC into two congruent right triangles.
Let's call the point where the altitude intersects side AC as D. This would make AD = CD, and AC would be the hypotenuse of one of the right triangles.
Using trigonometry, we can find the length of AC. In the right triangle with hypotenuse AC and angle DAB (which is half of angle ABC), we can use the sine function.
sin(DAB) = opposite/hypotenuse
sin(DAB) = BD/AB (opposite side / hypotenuse)
sin(60) = BD/6 (since AB = BC = 6 and angle ABC = 120)
√3 / 2 = BD/6
Cross-multiplying gives us:
2BD = 6√3
Simplifying, we get:
BD = 3√3
Since AD + CD = AC, and AD = CD = BD, we have:
AC = 2 * BD
AC = 2 * 3√3
AC = 6√3
Now, we have the value of AC as 6√3.
To find a + b, we need to find the values of a and b in the expression AC = √b.
From AC = 6√3, we can see that b = (6√3)^2.
Calculating that, we get:
b = (6 * 3)^2
b = (18)^2
b = 324
So, a + b = a + 324.
Without further information, we cannot determine the exact value of a. But we have concluded that a + b = a + 324.
To find the value of a+b, we need to determine the length of AC.
Given that ABC is an isosceles triangle with AB=BC=6, we can draw an altitude from vertex B to side AC to create a right triangle. Let's call the point of intersection D, where BD is the altitude.
Since ADC is a right triangle, we can use the Pythagorean theorem to find the length of AD or DC. By dividing triangle ABC into two right triangles, we can see that each right triangle is a 30-60-90 triangle.
In a 30-60-90 triangle, the side lengths are in the ratio of 1:√3:2. Therefore, in triangle BDC, the sides are BD = 6 (the height of the isosceles triangle), DC = BD * √3 (since DC is the longer leg in a 30-60-90 triangle), and BC = 6.
Using the Pythagorean theorem:
DC^2 = BC^2 - BD^2
(DC)^2 = 6^2 - 3^2
(DC)^2 = 36 - 9
(DC)^2 = 27
Taking the square root of both sides to find DC:
DC = √27
Now, we know that AC = AD + DC.
In the right triangle ADC, AD = AB = 6.
Therefore, AC = 6 + √27.
To simplify the square root, we note that 27 can be written as 9 * 3. Taking the square root of both, we have:
AC = 6 + √(9 * 3)
AC = 6 + 3√3
Thus, a = 6 and b = 3, so the sum of a and b is a + b = 6 + 3 = 9.