Question

22. Let [AB] be a chord in a circle C, and k a circle which is internally tangent to the circle C at a point P and to the chord [AB] at a point Q. Show that the line P Q passes through the midpoint of the arc AB opposite to the arc APB.

260

likes
1299 views

Answer to a math question 22. Let [AB] be a chord in a circle C, and k a circle which is internally tangent to the circle C at a point P and to the chord [AB] at a point Q. Show that the line P Q passes through the midpoint of the arc AB opposite to the arc APB.

Expert avatar
Dexter
4.7
112 Answers
Pour prouver que la droite PQ passe par le milieu de l’arc AB opposé à l’arc APB, on peut suivre ces étapes : 1. Soit O le centre du cercle C, et M le milieu de l'arc AB opposé à l'arc APB. 2. Puisque le cercle k est intérieurement tangent au cercle C au point P, nous savons que la droite OP est perpendiculaire à PQ. En effet, le rayon du cercle C au point P est perpendiculaire à toute ligne tangente passant par P. 3. Soit N le point d'intersection de la droite PQ et du cercle C. Il faut montrer que N est identique à M, milieu de l'arc AB opposé à l'arc APB. 4. Puisque OP est perpendiculaire à PQ, le triangle OPQ est un triangle rectangle. 5. Considérant le triangle rectangle OPQ, nous savons que l’hypoténuse OQ est un diamètre du cercle C. L’angle OQP est donc un angle droit. 6. Puisque l’angle OQP est un angle droit, et que l’angle OMP est également un angle droit (puisque M est le milieu de l’arc AB), on peut conclure que le quadrilatère OMNQ est un quadrilatère cyclique. 7. Par les propriétés d'un quadrilatère cyclique, les angles opposés de OMNQ sont supplémentaires. L’angle OMN est donc complémentaire à l’angle OQN. 8. Puisque l'angle OMN est supplémentaire à l'angle OQN, et que l'angle OQN est un angle droit, il s'ensuit que l'angle OMN est aussi un angle droit. 9. L'angle OMN étant un angle droit signifie que MN est perpendiculaire à la corde AB. 10. Puisque MN est perpendiculaire à la corde AB et que M est le milieu de l’arc AB opposé à l’arc APB, on peut conclure que la droite PQ passe par le milieu M de l’arc AB opposé à l’arc APB. Nous avons donc montré que la droite PQ passe par le milieu de l’arc AB opposé à l’arc APB.

Frequently asked questions (FAQs)
Math question: "Simplify the expression log(base 3) 27 - log(base 3) 9 + log(base 3) 81."
+
What is the result of multiplying vector A (2i - 3j) by vector B (5i + 4j)?
+
What is the average rate of change of the quadratic function f(x) = x^2 over the interval [0, 2]?
+
New questions in Mathematics
a to the power of 2 minus 16 over a plus 4, what is the result?
Revenue Maximization: A company sells products at a price of $50 per unit. The demand function is p = 100 - q, where p is the price and q is the quantity sold. How many units should they sell to maximize revenue?
what is 3% of 105?
Kayla has $8,836.00 in her savings account. The bank gives Kayla 5%of the amount of money in account as a customer bonus. What amount of money does the bank give Kayla? Justify your answer on a 6th grade level.
A bird randomly chooses to land on 1 of 12 perches available in its aviary. Determine the Probability of it landing on a perch numbered 8 and then on a perch marked with a prime number; take into account that he never lands on the same perch in the sequence.
∫ √9x + 1 dx
2x+4x=
A person decides to invest money in fixed income securities to redeem it at the end of 3 years. In this way, you make monthly deposits of R$300.00 in the 1st year, R$400.00 in the 2nd year and R$500.00 in the 3rd year. Calculate the amount, knowing that compound interest is 0.6% per month for the entire period. The answer is 15,828.60
Calculate the value of a so that the vectors (2,2,−1),(3,4,2) and(a,2,3) are coplanar.
A vaccine has a 90% probability of being effective in preventing a certain disease. The probability of getting the disease if a person is not vaccinated is 50%. In a certain geographic region, 60% of the people get vaccinated. If a person is selected at random from this region, find the probability that he or she will contract the disease. (4 Points)
Use a pattern to prove that (-2)-(-3)=1
Determine the increase of the function y=4x−5 when the argument changes from x1=2 to x2=3
9.25=2pi r solve for r
Write the detailed definition of a supply chain/logistics related maximization problem with 8 variables and 6 constraints. Each constraint should have at least 6 variables. Each constraint should have At least 5 variables will have a value greater than zero in the resulting solution. Variables may have decimal values. Type of equations is less than equal. Numbers and types of variables and constraints are important and strict. Model the problem and verify that is feasible, bounded and have at least 5 variables are nonzero.
A company dedicated to the manufacture of shirts sells the units at a price of $40, the cost of each shirt is $24, a commission is paid for the sale of a unit of shirt of $2 and its fixed costs are $3500 Determine the marginal contribution
25) Paulo saves R$250.00 per month and keeps the money in a safe in his own home. At the end of 12 months, deposit the total saved into the savings account. Consider that, each year, deposits are always carried out on the same day and month; the annual yield on the savings account is 7%; and, the yield total is obtained by the interest compounding process. So, the amount that Paulo will have in his savings account after 3 years, from the moment you started saving part of your money monthly, it will be A) R$6,644.70. B) R$ 9,210.00. C) R$ 9,644.70. D) R$ 10,319.83. E) R$ 13,319.83
The mean of 4 numbers is 5 and the mean of 3 different numbers is 12. What is the mean of the 7 numbers together? Produce an algebraic solution. Guess and check is acceptable.
Marc, Jean and Michelle have traveled a lot. Marc drove twice as much as Jean, but it was Michelle who drove the most with 100km more than Marc. They respected their objective of not exceeding 1350km of distance. How far did John drive?
Find the orthogonal projection of a point A = (1, 2, -1) onto a line passing through the points Pi = (0, 1, 1) and P2 = (1, 2, 3).
Beren spent 60% of the money in her piggy bank, and Ceren spent 7% of the money in her piggy bank to buy a joint gift for Deren, totaling 90 TL. In the end, it was observed that the remaining amounts in Ceren and Beren's piggy banks were equal. Therefore, what was the total amount of money that Beren and Ceren had initially? A) 120 B) 130 C) 150 D) 160 E) 180