Two chords AB and CD of A circle intersect at right angle at a point inside the circle if m(angle BAC is 35 degree find m(angle ABD)
frist 2019
125
Math question
Answer
To find the measure of angle ABD, we can use the properties of intersecting chords.
Step 1: Draw a diagram of the given situation. Label the points as described.
- Circle with center O
- Chords AB and CD intersecting at point E at a right angle.
- Angle BAC measures 35 degrees.
Step 2: Identify any relevant theorems or properties.
- The theorem we will use is that if two chords intersect in a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.
- In this case, we have AE * EB = CE * ED.
Step 3: Use the theorem to set up an equation that will allow us to solve for the unknown angle.
- Let AE = a and CE = b.
- Since AE * EB = CE * ED, we can substitute these values into the equation.
a * (a + EB) = b * (b + ED)
Step 4: Simplify the equation.
- Expand the equation by multiplying the terms.
a^2 + a * EB = b^2 + b * ED
Step 5: Use the information given to substitute known values into the equation.
- We know that angle BAC measures 35 degrees, and angle ABE is a right angle (90 degrees).
- This means angle AEB is 180 - 35 - 90 = 55 degrees.
- Substituting EB = a * sin(AEB) and ED = b * sin(AEC) into the equation, we get:
a^2 + a * a * sin(AEB) = b^2 + b * b * sin(AEC)
Step 6: Simplify the equation further using trigonometric identities.
- We can use the identities sin(AEB) = sin(90) = 1 and sin(AEC) = sin(AEB + 35) = sin(55 + 35) = sin(90) = 1.
a^2 + a^2 * 1 = b^2 + b^2 * 1
Step 7: Combine like terms and solve for the unknown quantities.
- Simplify the equation.
2a^2 = 2b^2
- Divide both sides by 2.
a^2 = b^2
- Take the square root of both sides.
a = b
Step 8: Determine the measure of angle ABD.
- Since AE = CE and angle AEB is a right angle, angle ABD and angle BCD will be congruent.
- Therefore, angle ABD measures 35 degrees.
In conclusion, the measure of angle ABD is 35 degrees.