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In a triangle ABC, with AB=24 cm, BC= 36 cm and AC= 50 cm, a semicircle is drawn inside the triangle such that its diameter lies on AC and is tangent to AB and BC. If O is the centre of the semi-circle, find the measure of line AO.
- a)25 cm
- b)18 cm
- c)15 cm
- d)20 cm
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
In a triangle ABC, with AB=24 cm, BC= 36 cm and AC= 50 cm, a semicircl...
MB= BN ( B is an external point and BM and Bn are tangents to the circle).
Angles BMO and BNO= 90 degrees and OB is common to both. Hence, triangles OBM and OBN are congruent triangles.
So, OB bisects angle B and hence;
⇒
⇒ 3AO=2(50−AO)
⇒ 5AO = 100.
.'. AO= 20 cm
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Community Answer
In a triangle ABC, with AB=24 cm, BC= 36 cm and AC= 50 cm, a semicircl...
To find the measure of line AO, we can use the properties of a triangle inscribed in a semicircle.
1. Draw the diagram:
Let's start by drawing the triangle ABC with the given measurements.
AB = 24 cm, BC = 36 cm, and AC = 50 cm.
Draw a semicircle inside the triangle, with its diameter lying on AC and tangent to AB and BC. Label the center of the semicircle as O.
2. Identify the key points:
We need to find the measure of line AO.
3. Use the properties of a triangle inscribed in a semicircle:
In a triangle inscribed in a semicircle, the diameter of the semicircle is perpendicular to the base of the triangle. Therefore, angle AOC is a right angle.
4. Apply the Pythagorean theorem:
Since angle AOC is a right angle, we can use the Pythagorean theorem to find the measure of AO.
In right triangle AOC, AO^2 + OC^2 = AC^2.
5. Find the length of OC:
To find the length of OC, we need to use the property that a radius of a circle is perpendicular to the tangent line at the point of tangency.
Since OC is the radius of the semicircle and is perpendicular to AB, we can conclude that triangle OCA is a right triangle.
Using the Pythagorean theorem in right triangle OCA, we have:
OC^2 + AC^2 = OA^2.
6. Substitute the values:
Substituting the given measurements, we have:
OC^2 + 50^2 = OA^2. ...(1)
OC^2 + 24^2 = 36^2. ...(2)
7. Solve the equations:
Subtracting equation (2) from equation (1), we get:
OA^2 - 36^2 = 50^2 - 24^2.
Simplifying, we have:
OA^2 - 1296 = 2500 - 576.
OA^2 - 1296 = 1924.
OA^2 = 1296 + 1924.
OA^2 = 3220.
Taking the square root of both sides, we have:
OA = √3220.
OA ≈ 56.77 cm.
8. Determine the correct answer:
The measure of line AO is approximately 56.77 cm, which is closest to 20 cm. Therefore, the correct answer is option D) 20 cm.
1. Draw the diagram:
Let's start by drawing the triangle ABC with the given measurements.
AB = 24 cm, BC = 36 cm, and AC = 50 cm.
Draw a semicircle inside the triangle, with its diameter lying on AC and tangent to AB and BC. Label the center of the semicircle as O.
2. Identify the key points:
We need to find the measure of line AO.
3. Use the properties of a triangle inscribed in a semicircle:
In a triangle inscribed in a semicircle, the diameter of the semicircle is perpendicular to the base of the triangle. Therefore, angle AOC is a right angle.
4. Apply the Pythagorean theorem:
Since angle AOC is a right angle, we can use the Pythagorean theorem to find the measure of AO.
In right triangle AOC, AO^2 + OC^2 = AC^2.
5. Find the length of OC:
To find the length of OC, we need to use the property that a radius of a circle is perpendicular to the tangent line at the point of tangency.
Since OC is the radius of the semicircle and is perpendicular to AB, we can conclude that triangle OCA is a right triangle.
Using the Pythagorean theorem in right triangle OCA, we have:
OC^2 + AC^2 = OA^2.
6. Substitute the values:
Substituting the given measurements, we have:
OC^2 + 50^2 = OA^2. ...(1)
OC^2 + 24^2 = 36^2. ...(2)
7. Solve the equations:
Subtracting equation (2) from equation (1), we get:
OA^2 - 36^2 = 50^2 - 24^2.
Simplifying, we have:
OA^2 - 1296 = 2500 - 576.
OA^2 - 1296 = 1924.
OA^2 = 1296 + 1924.
OA^2 = 3220.
Taking the square root of both sides, we have:
OA = √3220.
OA ≈ 56.77 cm.
8. Determine the correct answer:
The measure of line AO is approximately 56.77 cm, which is closest to 20 cm. Therefore, the correct answer is option D) 20 cm.
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