In the figure, ABC is a right and right angle at B. AD & CE arethe two medians drawn from A and C respectively. If AC = 5 cmand AD 3 5 cm.2 Then the length of CE will be(A) 4 5 cm (B) 2 5 cm(C) 3 5 cm2 (D) 6 5 cm
Question
In the figure, ABC is a right and right angle at B. AD & CE arethe two medians drawn from A and C respectively. If AC = 5 cmand AD 3 5 cm.2 Then the length of CE will be(A) 4 5 cm (B) 2 5 cm(C) 3 5 cm2 (D) 6 5 cm
Solution 1
Given that ΔABC is a right triangle with a right angle at B, and AD and CE are the two medians drawn from points A and C respectively, we are asked to find the length of CE.
We are given that AC = 5 cm and AD = 3/5 cm.
To find the length of CE, we can use the property of medians in a triangle. The length of a median divides the opposite side into two equal parts.
Since AD is a median, it divides BC into two equal parts. Therefore, BC = 2 * AD = 2 * (3/5) cm = 6/5 cm.
Similarly, CE is also a median, so it divides AB into two equal parts. Therefore, AB = 2 * CE.
Using the Pythagorean theorem in triangle ABC, we have:
AB^2 + BC^2 = AC^2
(2 * CE)^2 + (6/5)^2 = 5^2
4 * CE^2 + 36/25 = 25
4 * CE^2 = 25 - 36/25
4 * CE^2 = (625 - 36) / 25
4 * CE^2 = 589 / 25
CE^2 = 589 / (25 * 4)
CE^2 = 589 / 100
CE = √(589 / 100)
CE = √(589) / √(100)
CE = √(589) / 10
Therefore, the length of CE is √(589) / 10 cm.
Since none of the given answer choices match this value, it seems there may be an error in the question or the answer choices.
Solution 2
Given that ΔABC is a right triangle with a right angle at B, and AD and CE are the two medians drawn from points A and C respectively, we are asked to find the length of CE.
We are given that AC = 5 cm and AD = 3/5 cm.
To find the length of CE, we can use the property of medians in a triangle. The length of a median divides the opposite side into two equal parts.
Since AD is a median, it divides BC into two equal parts. Therefore, BC = 2 * AD = 2 * (3/5) cm = 6/5 cm.
Similarly, CE is also a median, so it divides AB into two equal parts. Therefore, AB = 2 * CE.
Using the Pythagorean theorem in triangle ABC, we have:
AB^2 + BC^2 = AC^2
(2 * CE)^2 + (6/5)^2 = 5^2
4 * CE^2 + 36/25 = 25
4 * CE^2 = 25 - 36/25
4 * CE^2 = (625 - 36) / 25
4 * CE^2 = 589 / 25
CE^2 = 589 / (25 * 4)
CE^2 = 589 / 100
CE = √(589 / 100)
CE = √(589) / √(100)
CE = √(589) / 10
Therefore, the length of CE is (√589) / 10 cm.
So, the correct answer is not provided in the options given.
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