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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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Verified Answer
In a triangle ABC, with AB = 24 cm, BC = 36 cm and AC = 50 cm, a semic...
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;
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
Most Upvoted Answer
In a triangle ABC, with AB = 24 cm, BC = 36 cm and AC = 50 cm, a semic...
Given:
In triangle ABC, AB = 24 cm, BC = 36 cm, and AC = 50 cm.
To find:
We need to find the measure of line AO.
Explanation:
To solve this question, we can use the property that the radius of a circle that is tangent to two sides of a triangle is perpendicular to the third side.
Let's draw the triangle ABC and the semicircle as described in the question.
Step 1: Drawing the triangle and semicircle
Draw a triangle ABC with sides AB = 24 cm, BC = 36 cm, and AC = 50 cm. Then, draw a semicircle inside the triangle such that its diameter lies on AC and is tangent to AB and BC. Let O be the center of the semicircle.
Step 2: Drawing the radii
Draw radii OA and OC from the center of the semicircle to the points A and C, respectively.
Step 3: Finding the length of OC
We know that OC is the radius of the semicircle. Since the semicircle is tangent to AB and BC, OC is perpendicular to AB and BC. Thus, OC is the height of the triangle ABC. We can find the length of OC using the formula for the area of a triangle:
Area of triangle ABC = 1/2 * base * height
Substituting the given values, we have:
1/2 * 36 * OC = 1/2 * 24 * OC
36 * OC = 24 * OC
OC = 24
Step 4: Finding the length of AO
Since O is the center of the semicircle, AO is the radius of the semicircle. Therefore, AO is equal to OC.
AO = OC = 24 cm
So, the measure of line AO is 24 cm.
Answer:
The measure of line AO is 24 cm. Therefore, the correct option is (d) 20 cm.
In triangle ABC, AB = 24 cm, BC = 36 cm, and AC = 50 cm.
To find:
We need to find the measure of line AO.
Explanation:
To solve this question, we can use the property that the radius of a circle that is tangent to two sides of a triangle is perpendicular to the third side.
Let's draw the triangle ABC and the semicircle as described in the question.
Step 1: Drawing the triangle and semicircle
Draw a triangle ABC with sides AB = 24 cm, BC = 36 cm, and AC = 50 cm. Then, draw a semicircle inside the triangle such that its diameter lies on AC and is tangent to AB and BC. Let O be the center of the semicircle.
Step 2: Drawing the radii
Draw radii OA and OC from the center of the semicircle to the points A and C, respectively.
Step 3: Finding the length of OC
We know that OC is the radius of the semicircle. Since the semicircle is tangent to AB and BC, OC is perpendicular to AB and BC. Thus, OC is the height of the triangle ABC. We can find the length of OC using the formula for the area of a triangle:
Area of triangle ABC = 1/2 * base * height
Substituting the given values, we have:
1/2 * 36 * OC = 1/2 * 24 * OC
36 * OC = 24 * OC
OC = 24
Step 4: Finding the length of AO
Since O is the center of the semicircle, AO is the radius of the semicircle. Therefore, AO is equal to OC.
AO = OC = 24 cm
So, the measure of line AO is 24 cm.
Answer:
The measure of line AO is 24 cm. Therefore, the correct option is (d) 20 cm.
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