answer:(a) Prove the criterion of equality of right triangles "by the two legs": 1. **Statement of the Criterion:** - We need to prove that if two right triangles have corresponding legs equal (i.e., the legs adjacent to the right angle), then these triangles are congruent. 2. **Triangle Configuration:** - Let triangle ABC and triangle A'B'C' be two right triangles with right angles at B and B' respectively. - Given that AB = A'B' and BC = B'C'. 3. **Application of Triangle Congruence Criteria:** - According to the **Pythagorean Theorem**, we can determine the hypotenuse for both triangles. - For triangle ABC, the hypotenuse AC is given by: [ AC = sqrt{AB^2 + BC^2} ] Substituting AB = A'B' and BC = B'C': [ AC = sqrt{A'B'^2 + B'C'^2} = sqrt{A'B'^2 + B'C'^2} = A'C' ] 4. **Triangle Congruence:** - Thus, we have three pairs of equal sides: AB = A'B', BC = B'C', and AC = A'C'. - By the **SSS (Side-Side-Side) criterion** for congruence, triangle ABC cong triangle A'B'C'. [ boxed{} ] (b) Formulate and prove the criterion of equality for right triangles by a leg and the adjacent acute angle: 1. **Statement of the Criterion:** - If a leg and an adjacent acute angle in one right triangle are equal to a leg and an adjacent acute angle in another right triangle, then the triangles are congruent. 2. **Triangle Configuration:** - Let triangle ABC and triangle A'B'C' be two right triangles with right angles at B and B' respectively. - Given that AB = A'B' and angle BAC = angle B'A'C'. 3. **Application of Congruence Criteria:** - Since triangles are right triangles, we know that angle ABC = angle A'B'C' = 90^circ. - Using the given information, angle BAC = angle B'A'C'. - The third angles, angle ACB and angle A'C'B' must be equal as the sum of angles in a triangle is 180^circ. 4. **Triangle Congruence by SAS (Side-Angle-Side):** - We have angle BAC = angle B'A'C', AB = A'B', and angle ABC = angle A'B'C' = 90^circ. - By the **SAS (Side-Angle-Side) criterion**, triangle ABC cong triangle A'B'C'. [ boxed{} ]

answer:If each of the 9 people paid approximately 16.99 after a 10% tip was added, we can calculate the total amount paid after the tip by multiplying 16.99 by 9. Total amount paid after tip = 16.99 * 9 Now, let's calculate the total amount paid after the tip: Total amount paid after tip = 16.99 * 9 = 152.91 (approximately) This total amount includes the original dining bill plus the 10% tip. To find the original dining bill before the tip, we need to subtract the tip from the total amount paid. Since the tip is 10% of the original dining bill, we can represent the original dining bill as "x" and the total amount paid as "1.10x" (which is 100% of the original bill plus 10% tip). So, we have: 1.10x = 152.91 To find the original dining bill (x), we divide the total amount paid after the tip by 1.10: x = 152.91 / 1.10 Now, let's calculate x: x = 152.91 / 1.10 = 139.01 (approximately) Therefore, the original dining bill amount before the tip was approximately boxed{139.01} .

answer:To determine the value of (Q) for a regular (Q)-sided polygon with (P) diagonals, follow these steps: 1. **Equation for the number of diagonals**: The number of diagonals in a polygon is determined by the formula: [ P = binom{Q}{2} - Q ] where (binom{Q}{2}) represents the number of ways to choose 2 vertices out of (Q) vertices, and we subtract (Q) because choosing two adjacent vertices results in sides of the polygon, not diagonals. 2. **Apply given information**: According to the problem statement, the number of diagonals (P) is given: [ P = 35 ] Therefore, we substitute (P) into the equation: [ 35 = binom{Q}{2} - Q ] 3. **Simplify the binomial coefficient**: Next, express the binomial coefficient (binom{Q}{2}) in its expanded form: [ binom{Q}{2} = frac{Q(Q-1)}{2} ] Substitute this into the equation: [ 35 = frac{Q(Q-1)}{2} - Q ] 4. **Clear the fraction**: Multiply the entire equation by 2 to clear the denominator: [ 2 cdot 35 = Q(Q-1) - 2Q ] Simplifying this gives: [ 70 = Q^2 - Q - 2Q ] Combine like terms: [ 70 = Q^2 - 3Q ] 5. **Rearrange into standard quadratic form**: Subtract 70 from both sides of the equation to set the equation to zero: [ Q^2 - 3Q - 70 = 0 ] 6. **Solve the quadratic equation**: Solve for (Q) using the quadratic formula (Q = frac{-b pm sqrt{b^2 - 4ac}}{2a}), where (a = 1), (b = -3), and (c = -70): [ Q = frac{-(-3) pm sqrt{(-3)^2 - 4 cdot 1 cdot (-70)}}{2 cdot 1} ] Simplify inside the square root: [ Q = frac{3 pm sqrt{9 + 280}}{2} ] [ Q = frac{3 pm sqrt{289}}{2} ] [ Q = frac{3 pm 17}{2} ] This gives two possible solutions: [ Q = frac{3 + 17}{2} = 10 ] [ Q = frac{3 - 17}{2} = -7 ] 7. **Interpret the solutions**: Since (Q) must be a positive integer (as it represents the number of sides of a polygon), we discard the negative solution (Q = -7) and accept (Q = 10). **Conclusion**: [ boxed{10} ]

answer:Problem: We have three categories of movie tickets, with prices 6 Ft, 4 Ft, and 3 Ft respectively for I., II., and III. seats. In the first year, the total revenue from these tickets forms 30%, 50%, and 20% of the overall revenue, respectively. These revenues saw changes in the second year: a 20% increase for I., a 30% increase for II., and a 5% decrease for III. seats. We are to determine how the total revenue changed and by how much the number of visitors to the cinema changed. Reference Solution Steps: 1. **First Year Revenue:** - Let the total revenue in the first year be (100A) Ft. - Revenue from I. tickets: (30A) Ft. - Revenue from II. tickets: (50A) Ft. - Revenue from III. tickets: (20A) Ft. 2. **Second Year Revenue Calculation:** - Revenue from I. tickets increases by 20%: [ 1.20 times 30A = 36A , text{Ft} ] - Revenue from II. tickets increases by 30%: [ 1.30 times 50A = 65A , text{Ft} ] - Revenue from III. tickets decreases by 5%: [ (1 - 0.05) times 20A = 0.95 times 20A = 19A , text{Ft} ] - Total revenue in the second year: [ 36A + 65A + 19A = 120A , text{Ft} ] - Thus, the total revenue increases by: [ 120A - 100A = 20A , text{Ft} quad text{which is a 20% increase}. ] 3. **Number of Tickets Sold in First Year:** - Number of I. tickets sold: [ frac{30A}{6} = 5A ] - Number of II. tickets sold: [ frac{50A}{4} = 12.5A ] - Number of III. tickets sold: [ frac{20A}{3} approx 6.67A ] - Total number of tickets: [ 5A + 12.5A + 6.67A = frac{145A}{6} ] 4. **Number of Tickets Sold in Second Year:** - Number of I. tickets sold: [ frac{36A}{6} = 6A ] - Number of II. tickets sold: [ frac{65A}{4} = 16.25A ] - Number of III. tickets sold: [ frac{19A}{3} approx 6.33A ] - Total number of tickets: [ 6A + 16.25A + 6.33A = frac{171.5A}{6} ] 5. **Percentage Increase in Number of Tickets Sold:** - The total for the second year was (frac{171.5A}{6}), and the total for the first year was (frac{145A}{6}). - The ratio of the second year total to the first year total: [ frac{frac{171.5A}{6}}{frac{145A}{6}} = frac{171.5}{145} approx 1.1827 ] - Therefore, the number of visitors increased by approximately: [ (1.1827 - 1) times 100% = 0.1827 times 100% = 18.27% ] Conclusion: The total revenue increased by 20%, and the number of cinema visitors increased by 18.27%. [ boxed{20%, 18.27%} ]