Hi, could someone explain to me the steps I should take to solve the following problem. If radius BE is a perpendicular bisector of chord AC, then what is AB to the nearest tenth? BD is 7cm and DC is 9.
You are obviously looking at some circle diagram that I can't see.
In BE , which is the centre, B or E ?
Where does DC come from?
Where does BD come from ?
All I know so far is that AC is a chord.
One wonders where point D is.
To solve this problem, we can use the properties of a perpendicular bisector. Follow these steps:
1. Draw a diagram: Draw a circle with center B and a chord AC. Mark the midpoint of AC as M. Draw radius BE perpendicular to AC.
2. Identify what is given: The length of BD is given as 7 cm and the length of DC is given as 9 cm.
3. Determine the lengths: Since BE is a perpendicular bisector of AC, we know that AM = MC. Let's represent AB as x, then MB also equals x. We can use the Pythagorean theorem to find values for x.
- In right triangle ABD, we have BD = 7 cm, and AB = x cm.
- In right triangle BDC, we have DC = 9 cm, and BC = (AB + BD) cm = (x + 7) cm.
4. Apply the Pythagorean theorem: In right triangle ABD, we have the equation (AB)^2 + (BD)^2 = (AD)^2. Likewise, in right triangle BDC, we have the equation (BC)^2 + (DC)^2 = (BD)^2.
- In triangle ABD: (x)^2 + (7)^2 = (AD)^2.
- In triangle BDC: (x + 7)^2 + (9)^2 = (BD)^2.
5. Solve the equations: We now have two equations with two unknowns, (AD)^2 and (BD)^2. Simplify and solve the system of equations to find the value of x.
- Simplify the equation in triangle ABD: x^2 + 49 = (AD)^2.
- Simplify the equation in triangle BDC: (x + 7)^2 + 81 = 49.
Solving the second equation gives x^2 + 14x + 49 + 81 - 49 = 0,
Simplifying further gives x^2 + 14x + 81 = 0.
Notice that this is a quadratic equation. We can solve it using factoring, completing the square, or using the quadratic formula.
By using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
= (-14 ± √(14^2 - 4(1)(81))) / 2(1)
= (-14 ± √(196 - 324)) / 2
= (-14 ± √(-128)) / 2
= (-14 ± 11.31i) / 2
= -7 ± 5.66i.
Since we are looking for a real value for x, we can ignore the imaginary solutions.
Therefore, x = -7 + 5.66i or x = -7 - 5.66i.
However, in this case, we are only interested in the side AB, which represents a length. Therefore, we take the positive real solution, x ≈ -7 + 5.66i, and ignore the negative value.
Hence, AB ≈ -7 + 5.66i.
6. Calculate the value of AB: Taking the positive real solution from step 5, we find that AB ≈ -7 + 5.66i cm.
To summarize, the value of AB is approximately -7 + 5.66i cm.