Given an isosceles triangle ABC where AB=AC and the base BC is of length 8 cm. The height from A to the base BC divides it into two segments, BD and DC. If the length of BD is 3 cm, what is the length of segment DC and the height of the triangle?

 Apr 11, 2024

We can solve this problem using the properties of isosceles triangles and the Pythagorean theorem.


Isosceles Triangle Properties:


Since AB = AC in an isosceles triangle ABC, angles B and C are congruent.


Dividing the Base:


The height from A to BC is the perpendicular bisector of the base (divides it into two segments of equal length) in an isosceles triangle.


Segment Lengths:


Given that BC = 8 cm and BD = 3 cm, we know DC = BC - BD = 8 cm - 3 cm = 5 cm.


Finding the Height:


We can use the Pythagorean theorem on right triangle ABD (since the height is perpendicular to the base).


Pythagorean Theorem:


a^2 + b^2 = c^2 (where a and b are the lengths of the legs and c is the length of the hypotenuse)


In this case:


a (length of leg BD) = 3 cm


c (length of hypotenuse AB) = unknown (represents the height of the triangle)


b (length of leg AD) = unknown


Solving for Height (c):


b^2 = c^2 - a^2 (rearranging the equation)


b^2 = c^2 - 3^2 (substitute known values)


Since AD is half of the base (property of isosceles triangle):


AD = 1/2 * BC = 1/2 * 8 cm = 4 cm


Substitute the value of AD (b) in the equation:


4^2 = c^2 - 3^2




16 = c^2 - 9


Add 9 to both sides:


25 = c^2


Take the square root of both sides (remember to consider both positive and negative since squaring either results in 25):


c = ± 5


Positive Height:


Since the height represents a distance, we take the positive value:


c (height of the triangle) = 5 cm




Length of segment DC = 5 cm


Height of the triangle = 5 cm

 Apr 14, 2024

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