C lies on a circle with diameter AD. B lies on AC and E lies on AD such that BE is parallel to CD. AB=21 cm, CD=18 cm and BE=13.5 cm. (a) Show that triangle ABE is similar to triangle ACD. (b) Find the length of the diameter AD. (c) Work out the circumference of the circle.

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Asked on July 9, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-07-09T13:10:56.287Z

Here are the solutions to the problems from the image: 14. (a) Show that triangle ABE is similar to triangle ACD. Step 1: Identify common angles. Both triangles ABE and ACD share the angle at vertex A. BAE = CAD (Common angle) Step 2: Identify angles formed by parallel lines. Given that BE is parallel to CD (BE CD). Since AC is a transversal line intersecting two parallel lines, the corresponding angles are equal. ABE = ACD (Corresponding angles) Alternatively, since AD is a transversal line intersecting two parallel lines, the corresponding angles are equal. AEB = ADC (Corresponding angles) Step 3: Conclude similarity. Since two pairs of corresponding angles are equal ( BAE = CAD and ABE = ACD), the triangles are similar by the Angle-Angle (AA) similarity criterion. Therefore, ABE ACD. 14. (b) Find the length of the diameter AD. Step 1: Use the similarity of triangles to find AC. Since ABE ACD, the ratio of corresponding sides is equal: (AB)/(AC) = (BE)/(CD) Substitute the given values: AB = 21 cm, BE = 13.5 cm, CD = 18 cm. (21)/(AC) = (13.5)/(18) To find AC, rearrange the equation: AC = (21 × 18)/(13.5) AC = (378)/(13.5) AC = 28 cm Step 2: Use the property of angles in a semicircle. The problem states that C lies on a circle with diameter AD. An angle subtended by a diameter at any point on the circumference is a right angle (90^). Therefore, ACD = 90^. This means ACD is a right-angled triangle with hypotenuse AD. Step 3: Use the Pythagorean theorem to find AD. In right-angled ACD: AD^2 = AC^2 + CD^2 Substitute the values AC = 28 cm and CD = 18 cm: AD^2 = (28)^2 + (18)^2 AD^2 = 784 + 324 AD^2 = 1108 AD = sqrt(1108) AD ≈ 33.2866 cm The length of the diameter AD is 33.3 cm (rounded to 3 significant figures). 14. (c) Work out the circumference of the circle. Step 1: Use the formula for the circumference of a circle. The circumference C of a circle is given by C = d, where d is the diameter. From part (b), the diameter d = AD = sqrt(1108) cm. Step 2: Calculate the circumference. C = × sqrt(1108) C ≈ 3.14159 × 33.2866 C ≈ 104.564 cm Rounding to one decimal place: C ≈ 104.6 cm The circumference of the circle is 104.6 cm. What's next?

Accepted Answer · 2026-07-09T13:10:56.287Z

Here are the solutions to the problems from the image: 14. (a) Show that triangle ABE is similar to triangle ACD. Step 1: Identify common angles. Both triangles ABE and ACD share the angle at vertex A. BAE = CAD (Common angle) Step 2: Identify angles formed by parallel lines. Given that BE is parallel to CD (BE CD). Since AC is a transversal line intersecting two parallel lines, the corresponding angles are equal. ABE = ACD (Corresponding angles) Alternatively, since AD is a transversal line intersecting two parallel lines, the corresponding angles are equal. AEB = ADC (Corresponding angles) Step 3: Conclude similarity. Since two pairs of corresponding angles are equal ( BAE = CAD and ABE = ACD), the triangles are similar by the Angle-Angle (AA) similarity criterion. Therefore, ABE ACD. 14. (b) Find the length of the diameter AD. Step 1: Use the similarity of triangles to find AC. Since ABE ACD, the ratio of corresponding sides is equal: (AB)/(AC) = (BE)/(CD) Substitute the given values: AB = 21 cm, BE = 13.5 cm, CD = 18 cm. (21)/(AC) = (13.5)/(18) To find AC, rearrange the equation: AC = (21 × 18)/(13.5) AC = (378)/(13.5) AC = 28 cm Step 2: Use the property of angles in a semicircle. The problem states that C lies on a circle with diameter AD. An angle subtended by a diameter at any point on the circumference is a right angle (90^). Therefore, ACD = 90^. This means ACD is a right-angled triangle with hypotenuse AD. Step 3: Use the Pythagorean theorem to find AD. In right-angled ACD: AD^2 = AC^2 + CD^2 Substitute the values AC = 28 cm and CD = 18 cm: AD^2 = (28)^2 + (18)^2 AD^2 = 784 + 324 AD^2 = 1108 AD = sqrt(1108) AD ≈ 33.2866 cm The length of the diameter AD is 33.3 cm (rounded to 3 significant figures). 14. (c) Work out the circumference of the circle. Step 1: Use the formula for the circumference of a circle. The circumference C of a circle is given by C = d, where d is the diameter. From part (b), the diameter d = AD = sqrt(1108) cm. Step 2: Calculate the circumference. C = × sqrt(1108) C ≈ 3.14159 × 33.2866 C ≈ 104.564 cm Rounding to one decimal place: C ≈ 104.6 cm The circumference of the circle is 104.6 cm. What's next?

C lies on a circle with diameter AD. B lies on AC and E lies on AD such that BE is parallel to CD. AB=21 cm, CD=18 cm and BE=13.5 cm. (a) Show that triangle ABE is similar to triangle ACD. (b) Find the length of the diameter AD. (c) Work out the circumference of the circle.

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C lies on a circle with diameter AD. B lies on AC and E lies on AD such that BE is parallel to CD. AB=21 cm, CD=18 cm and BE=13.5 cm. (a) Show that triangle ABE is similar to triangle ACD. (b) Find the length of the diameter AD. (c) Work out the circumference of the circle.

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