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ΔABC has altitudes AD, BE and CF. Which of the following is/are true?
A. The ratio of the perimeter of the triangle to the sum of its altitudes is greater than 1.
B. Each altitude of the triangle is less than half the perimeter of the triangle.
A. The ratio of the perimeter of the triangle to the sum of its altitudes is greater than 1.
B. Each altitude of the triangle is less than half the perimeter of the triangle.
- a)Only statement A
- b)Only statement B
- c)Neither statement A nor statement B
- d)Both statement A and statement B
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
ABC has altitudes AD, BE and CF. Which of the following is/are true?A....
AD, BE and CF are altitudes of ΔABC.
Since, the perpendicular is the shortest distance between a point (A) and a line (BC),
AD < AB and AD < AC
Adding the two inequations,
2AD < (AB + AC) ...(i)
Similarly,
2BE < (AB + BC) ...(ii)
and
2CF < (AC + BC) _(iii)
Adding (i), (ii) and (iii),
(AD + BE + CF) < (AB + BC + AC)
(AB + BC + AC)/(AD + BE + CF) > 1
Statement A is true.
Also, from (i)
2AD < (AB + BC) < (AB + BC + AC)
Similarly,
2BE < AB + BC + AC
2CF < AB + BC + AC
Statement B is true.
Hence, option 4.
Most Upvoted Answer
ABC has altitudes AD, BE and CF. Which of the following is/are true?A....
Statement A: The ratio of the perimeter of the triangle to the sum of its altitudes is greater than 1.
Statement B: Each altitude of the triangle is less than half the perimeter of the triangle.
Explanation:
To understand the given statements, let's consider a triangle ABC with altitudes AD, BE, and CF.
Statement A: The ratio of the perimeter of the triangle to the sum of its altitudes is greater than 1.
To prove this statement, we need to compare the perimeter of the triangle (AB + BC + CA) with the sum of its altitudes (AD + BE + CF).
Let's assume the lengths of AB, BC, and CA are a, b, and c respectively. Now, let's calculate the sum of the altitudes.
The area of a triangle is given by the formula: Area = 1/2 * base * height.
Since AD is the altitude corresponding to side BC, the area of triangle ADB is 1/2 * BC * AD.
Similarly, the area of triangle AEC is 1/2 * CA * BE, and the area of triangle BFC is 1/2 * AB * CF.
The sum of the altitudes is given by: AD + BE + CF = 2 * (Area of ADB + Area of AEC + Area of BFC)
= BC * AD + CA * BE + AB * CF.
Now, let's calculate the perimeter of the triangle.
Perimeter = AB + BC + CA = a + b + c.
To compare the perimeter with the sum of the altitudes, we need to find the relationship between the altitudes and the sides of the triangle.
Statement B: Each altitude of the triangle is less than half the perimeter of the triangle.
To prove this statement, we need to show that AD < a/2,="" be="" />< b/2,="" and="" cf="" />< />
Let's consider the altitude AD. Since AD is perpendicular to BC, it divides the triangle into two right-angled triangles. Let's call the points where AD intersects BC as M.
In triangle ABD, the hypotenuse is AB and the altitude is AD. By applying the Pythagorean theorem, we can write:
AB^2 = AM^2 + BM^2.
Similarly, in triangle ACD, the hypotenuse is AC and the altitude is AD. By applying the Pythagorean theorem, we can write:
AC^2 = (BC - BM)^2 + AD^2.
By adding the above two equations, we get:
AB^2 + AC^2 = AM^2 + BM^2 + (BC - BM)^2 + AD^2.
Simplifying the equation, we get:
AB^2 + AC^2 = 2(AM^2 + AD^2).
Since AM^2 + AD^2 is always positive, AB^2 + AC^2 > 2AD^2.
By taking the square root on both sides, we get:
sqrt(AB^2 + AC^2) > sqrt(2) * AD.
Now, let's consider the perimeter of the triangle and the sum of its altitudes.
Perimeter = AB + BC + CA = a + b + c.
Sum of altitudes = AD + BE + CF.
Statement B: Each altitude of the triangle is less than half the perimeter of the triangle.
Explanation:
To understand the given statements, let's consider a triangle ABC with altitudes AD, BE, and CF.
Statement A: The ratio of the perimeter of the triangle to the sum of its altitudes is greater than 1.
To prove this statement, we need to compare the perimeter of the triangle (AB + BC + CA) with the sum of its altitudes (AD + BE + CF).
Let's assume the lengths of AB, BC, and CA are a, b, and c respectively. Now, let's calculate the sum of the altitudes.
The area of a triangle is given by the formula: Area = 1/2 * base * height.
Since AD is the altitude corresponding to side BC, the area of triangle ADB is 1/2 * BC * AD.
Similarly, the area of triangle AEC is 1/2 * CA * BE, and the area of triangle BFC is 1/2 * AB * CF.
The sum of the altitudes is given by: AD + BE + CF = 2 * (Area of ADB + Area of AEC + Area of BFC)
= BC * AD + CA * BE + AB * CF.
Now, let's calculate the perimeter of the triangle.
Perimeter = AB + BC + CA = a + b + c.
To compare the perimeter with the sum of the altitudes, we need to find the relationship between the altitudes and the sides of the triangle.
Statement B: Each altitude of the triangle is less than half the perimeter of the triangle.
To prove this statement, we need to show that AD < a/2,="" be="" />< b/2,="" and="" cf="" />< />
Let's consider the altitude AD. Since AD is perpendicular to BC, it divides the triangle into two right-angled triangles. Let's call the points where AD intersects BC as M.
In triangle ABD, the hypotenuse is AB and the altitude is AD. By applying the Pythagorean theorem, we can write:
AB^2 = AM^2 + BM^2.
Similarly, in triangle ACD, the hypotenuse is AC and the altitude is AD. By applying the Pythagorean theorem, we can write:
AC^2 = (BC - BM)^2 + AD^2.
By adding the above two equations, we get:
AB^2 + AC^2 = AM^2 + BM^2 + (BC - BM)^2 + AD^2.
Simplifying the equation, we get:
AB^2 + AC^2 = 2(AM^2 + AD^2).
Since AM^2 + AD^2 is always positive, AB^2 + AC^2 > 2AD^2.
By taking the square root on both sides, we get:
sqrt(AB^2 + AC^2) > sqrt(2) * AD.
Now, let's consider the perimeter of the triangle and the sum of its altitudes.
Perimeter = AB + BC + CA = a + b + c.
Sum of altitudes = AD + BE + CF.
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Community Answer
ABC has altitudes AD, BE and CF. Which of the following is/are true?A....
Altitude makes an angle of 90 degrees at the side of the triangle so side opposite to altitude ie. side of the triangle will be the hypotenuse of the smaller triangle.
therefore all the three sides will be greater than their altitude as hypotenuse js greater than height in Pythagoras
1. so the ratio of the perimeter will be greater than 1
2. and altitude will be less than half of perimeter
therefore all the three sides will be greater than their altitude as hypotenuse js greater than height in Pythagoras
1. so the ratio of the perimeter will be greater than 1
2. and altitude will be less than half of perimeter
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ABC has altitudes AD, BE and CF. Which of the following is/are true?A. The ratio of the perimeter of the triangle to the sum of its altitudes is greater than 1.B. Each altitude of the triangle is less than half the perimeter of the triangle.a)Only statement Ab)Only statement Bc)Neither statement A nor statement Bd)Both statement A and statement BCorrect answer is option 'D'. Can you explain this answer?
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